Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 612304, 12 pages http://dx.doi.org/10.1155/2014/612304

Research Article Homomorphisms between Algebras of Holomorphic Functions Verónica Dimant,1 Domingo García,2 Manuel Maestre,2 and Pablo Sevilla-Peris3 1

Departamento de Matem´atica, Universidad de San Andr´es, Vito Dumas 284, Victoria, B1644BID Buenos Aires, Argentina Departamento de An´alisis Matem´atico, Universidad de Valencia, Doctor Moliner 50, Burjasot, 46100 Valencia, Spain 3 Instituto Universitario de Matem´atica Pura y Aplicada, Universitat Polit`ecnica de Val`encia, 46022 Valencia, Spain 2

Correspondence should be addressed to Pablo Sevilla-Peris; [email protected] Received 26 December 2013; Accepted 13 March 2014; Published 12 May 2014 Academic Editor: Alfredo Peris Copyright © 2014 Ver´onica Dimant et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For two complex Banach spaces 𝑋 and 𝑌, in this paper, we study the generalized spectrum M𝑏 (𝑋, 𝑌) of all nonzero algebra homomorphisms from H𝑏 (𝑋), the algebra of all bounded type entire functions on 𝑋, into H𝑏 (𝑌). We endow M𝑏 (𝑋, 𝑌) with a structure of Riemann domain over L(𝑋∗ , 𝑌∗ ) whenever 𝑋 is symmetrically regular. The size of the fibers is also studied. Following the philosophy of (Aron et al., 1991), this is a step to study the set M𝑏,∞ (𝑋, 𝐵𝑌 ) of all nonzero algebra homomorphisms from H𝑏 (𝑋) into H∞ (𝐵𝑌 ) of bounded holomorphic functions on the open unit ball of 𝑌 and M∞ (𝐵𝑋 , 𝐵𝑌 ) of all nonzero algebra homomorphisms from H∞ (𝐵𝑋 ) into H∞ (𝐵𝑌 ).

1. Introduction The study of homomorphisms between topological algebras is one of the basic issues in this theory. Two are the main topological algebras that we come across when we deal with holomorphic functions on infinite dimensional spaces (see Section 2 for precise definitions): H𝑏 (𝑋), the holomorphic functions of bounded type (which is a Fr´echet algebra), and H∞ (𝐵𝑋 ), the bounded holomorphic functions on the open unit ball (which is a Banach algebra). Here, as a first step in the study of the set of homomorphisms between H∞ (𝐵𝑋 ) spaces, we mainly focus on algebras of holomorphic functions of bounded type and homomorphisms between them; 𝐿 : H𝑏 (𝑋) → H𝑏 (𝑌) (i.e., continuous, linear, and multiplicative mappings). These were already considered in [1]. There the focus was to study the homomorphisms as “individuals,” seeking properties of single ones. We have here a different interest: we treat them as a whole, considering the set

M𝑏 (𝑋, 𝑌) = M (H𝑏 (𝑋) , H𝑏 (𝑌)) = {Φ : H𝑏 (𝑋) → H𝑏 (𝑌) algebra homomorphisms} \ {0} .

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We will call this set the generalized spectrum or simply the spectrum. Our main aim is to study M𝑏 (𝑋, 𝑌) and to define on it a topological and a differential structure. This problem has the same flavor as considering M𝑏 (𝑋), the spectrum of the algebra H𝑏 (𝑋) (i.e., the set of nonzero continuous, linear, and multiplicative Φ : H𝑏 (𝑋) → C). This was studied in [2, 3], where a structure of Riemannian manifold over the bidual 𝑋∗∗ was defined on it (see also [4, Section 3.6] for a very neat and nice presentation and [5–7] for similar results). Our approach is very much indebted to that in [2] and we get up to some point analogous results, defining on M𝑏 (𝑋, 𝑌) a Riemann structure over L(𝑋∗ , 𝑌∗ ) (note that 𝑋∗∗ = L(𝑋∗ , C)). We will also be interested in the fibers of elements of L(𝑋∗ , 𝑌∗ ). The outline of the paper is the following. In Section 3, for two complex Banach spaces 𝑋 and 𝑌, we study the generalized spectrum M𝑏 (𝑋, 𝑌) of all nonzero algebra homomorphisms from H𝑏 (𝑋) to H𝑏 (𝑌). We endow it with a structure of Riemann domain over L(𝑋∗ , 𝑌∗ ) whenever 𝑋 is symmetrically regular. In Section 4, we focus on the sets (fibers) of elements in M𝑏 (𝑋, 𝑌) that are projected on the same element 𝑢 of L(𝑋∗ , 𝑌∗ ). The size of these fibers is studied and we prove that they are big by showing that they contain big sets. Following the philosophy of [2], all about M𝑏 (𝑋, 𝑌) is a step to study in Section 5 the spectrum

2

Abstract and Applied Analysis

M𝑏,∞ (𝑋, 𝐵𝑌 ) of all nonzero algebra homomorphisms from H𝑏 (𝑋) to H∞ (𝐵𝑌 ) of bounded holomorphic functions on the open unit ball of 𝑌. Finally, in Section 6, we deal with the generalized spectrum M∞ (𝐵𝑋 , 𝐵𝑌 ) of all nonzero algebra homomorphisms from H∞ (𝐵𝑋 ) to H∞ (𝐵𝑌 ).

2. Definitions and Preliminaries Unless otherwise stated capital letters such as 𝑋, 𝑌, . . . will denote complex Banach spaces. The dual will be denoted by 𝑋∗ and the open ball of center 𝑥0 and radius 𝑟 > 0 by 𝐵𝑋 (𝑥0 , 𝑟). If 𝑥0 = 0 and 𝑟 = 1 we just write 𝐵𝑋 . The space of continuous, linear operators from 𝑋 to 𝑌 will be denoted by L(𝑋, 𝑌); this is a Banach space with the norm ‖𝑢‖ = sup𝑥∈𝐵𝑋 ‖𝑢(𝑥)‖. The adjoint operator of 𝑢 ∈ L(𝑋, 𝑌) will be denoted by 𝑢∗ ∈ L(𝑌∗ , 𝑋∗ ). We will denote the canonical inclusion of a space into its bidual by 𝐽𝑋 : 𝑋 → 𝑋∗∗ . Given two complex locally convex spaces 𝐸 and 𝐹, a mapping 𝑃 : 𝐸 → 𝐹 is a continuous 𝑚-homogeneous polynomial if there is a continuous 𝑚-linear mapping 𝐿 : 𝐸 × ⋅ ⋅ ⋅ × 𝐸 → 𝐹 such that 𝑃 (𝑥) = 𝐿 (𝑥, . . . , 𝑥)

for every 𝑥.

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Throughout the paper every polynomial and multilinear mapping will be assumed to be continuous. An 𝑚-linear mapping 𝐿 is called symmetric if 𝐿(𝑥𝜎1 , . . . , 𝑥𝜎𝑚 ) = 𝐿(𝑥1 , . . . , 𝑥𝑚 ) for every permutation 𝜎 of {1, . . . , 𝑚}. Each 𝑚-homogeneous polynomial has a unique ∨

symmetric mapping (which we denote by 𝑃) satisfying (2). If 𝑃 : 𝑋 → 𝑌 is a continuous 𝑚-homogeneous polynomial between Banach spaces, the following expressions define norms for 𝑚-linear mappings and for 𝑚-homogeneous polynomials, respectively: ‖𝐿‖ = sup {𝐿 (𝑥1 , . . . , 𝑥𝑚 ) : 𝑥𝑗 ≤ 1, 𝑗 = 1, . . . , 𝑚} , ‖𝑃‖ = sup {‖𝑃 (𝑥)‖ : ‖𝑥‖ ≤ 1} .

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Given an 𝑚-linear mapping 𝐿, the notation 𝐿(𝑥(𝑘) , 𝑦(𝑚−𝑘) ) will mean that 𝑥 is repeated 𝑘 times and 𝑦 is repeated 𝑚 − 𝑘 times. A mapping 𝑓 : 𝑈 ⊆ 𝐸 → 𝐹 is holomorphic on the open set 𝑈 if for every 𝑥0 in 𝑈, there exist (𝑃𝑚 𝑓(𝑥0 ))𝑚 , each one of them an 𝑚-homogeneous polynomial, such that the series ∞

𝑓 (𝑥) = ∑ 𝑃𝑚 𝑓 (𝑥0 ) (𝑥 − 𝑥0 )

A holomorphic function 𝑓 : 𝐸 → 𝐹 is of bounded type if it sends bounded subsets of 𝐸 to bounded sets of 𝐹. We denote by H𝑏 (𝐸, 𝐹) the space of holomorphic functions of bounded type from 𝐸 to 𝐹. If 𝐹 = C we simply write H𝑏 (𝐸). Given 𝑈 ⊆ 𝐸 we write 𝑓𝑈 = sup 𝑓 (𝑥) . (7) 𝑥∈𝑈

With this notation for 𝑋 a Banach space H𝑏 (𝑋) is a Fr´echet algebra endowed with the topology 𝜏𝑏 of uniform convergence on the bounded sets, whose seminorms are (for 𝑅 > 0) 𝑞𝑅 (𝑓) = 𝑓𝐵𝑋 (0,𝑅) = sup {𝑓 (𝑥) : ‖𝑥‖ < 𝑅} . (8) Let 𝑀 be a differential manifold on a complex Banach space 𝑋 and 𝐹 a locally convex space. A mapping 𝑓 : 𝑀 → 𝐹 is said to be holomorphic (of bounded type) if 𝑓 ∘ 𝜑−1 : Ω → 𝐹 is holomorphic (of bounded type) for every chart (𝜑, Ω) of 𝑀. Given 𝑥 ∈ 𝑋, we write 𝛿𝑥 for the evaluation mapping at 𝑥; that is, 𝛿𝑥 (𝑓) = 𝑓(𝑥) for all holomorphic 𝑓. Let 𝐿 : 𝑋×𝑋 → C be a continuous bilinear form. Fix 𝑥 ∈ 𝑋 and for 𝑤 ∈ 𝑋∗∗ let (𝑦𝛽 ) be a net in 𝑋 weak-star convergent to 𝑤. Since 𝐿(𝑥, −) ∈ 𝑋∗ , then there exists lim𝛽 𝐿(𝑥, 𝑦𝛽 ) := 𝐿(𝑥, 𝑤). Now, fix 𝑤 ∈ 𝑋∗∗ and for 𝑧 ∈ 𝑋∗∗ let (𝑥𝛼 ) be a net in 𝑋 weak-star convergent to 𝑧. Since 𝐿(−, 𝑤) ∈ 𝑋∗ , then there exists ̃ (𝑧, 𝑤) := lim 𝐿 (𝑥𝛼 , 𝑤) = lim lim 𝐿 (𝑥𝛼 , 𝑦𝛽 ) . 𝐿 𝛼 𝛼 𝛽

The polarization formula gives [4, Corollary 1.8] ∨ 𝑚𝑚 ‖𝑃‖ ≤ 𝑃 ≤ ‖𝑃‖ . 𝑚!

known [4, page 152-153] that 𝑓 is holomorphic if and only if it is Gˆateaux holomorphic and continuous. In the case that 𝑓 : 𝑋 → 𝑌, 𝑋 and 𝑌 being complex Banach spaces, 𝑓 is holomorphic on 𝑋 if and only if it is Fr´echet differentiable on 𝑋 and in that case 𝑃1 (𝑥) = 𝑑𝑓(𝑥) for every 𝑥. Also, by the Cauchy inequalities [4, Proposition 3.2] we have, for every 𝑚, sup 𝑃𝑚 (𝑥0 ) (𝑥) ≤ sup 𝑓 (𝑥) . (6) 𝑥∈𝐵𝑋 (0,𝑅) 𝑥∈𝐵𝑋 (𝑥0 ,𝑅)

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𝑚=0

converges uniformly in some neighborhood of 𝑥0 contained in 𝑈. This is called the “Taylor series expansion” of 𝑓 at 𝑥0 . If 𝐸 and 𝐹 are Fr´echet spaces, then 𝑓 : 𝐸 → 𝐹 is Gˆateaux holomorphic if for every 𝑥, 𝑦 ∈ 𝐸 the function C → 𝐹, 𝑡 → 𝑓(𝑥 + 𝑡𝑦) is holomorphic on some neighborhood of 0. It is

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Following this idea Aron and Berner showed in [8] that every function 𝑓 ∈ H𝑏 (𝑋) admits an extension to the bidual, called the Aron-Berner extension, 𝑓 ∈ H𝑏 (𝑋∗∗ ). By [9, Theorem 3], for every 𝑚-homogeneous polynomial 𝑃, we have ‖𝑃‖ = ‖𝑃‖. A Banach space 𝑋 is symmetrically regular if for all continuous symmetric bilinear form 𝐿 : 𝑋 × 𝑋 → C it follows that ̃ (𝑧, 𝑤) = lim lim 𝐿 (𝑥𝛼 , 𝑦𝛽 ) . 𝐿 𝛼 𝛽

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We refer the reader to [4, 10, 11] for the general background on the theory of holomorphic functions on infinite dimensional spaces. We are going to work with the set M𝑏 (𝑋, 𝑌) of nonzero algebra homomorphisms between spaces of holomorphic functions of bounded type defined in (1). Observe that an idempotent element 𝑔 in H𝑏 (𝑌) satisfies that 𝑔(𝑌) is a subset of {0, 1} and 𝑔(𝑌) is a connected set. Hence, either 𝑔 ≡ 0 or 𝑔 ≡ 1. So, for any Φ ∈ M𝑏 (𝑋, 𝑌), we should have Φ(1𝑋 ) = 1𝑌 .

Abstract and Applied Analysis

3

3. The Differential Structure of M𝑏 (𝑋,𝑌)

for all 𝑔 ∈ H𝑏 (𝑋). Hence

Our aim in this section is to endow M𝑏 (𝑋, 𝑌) with a structure of Riemann domain over L(𝑋∗ , 𝑌∗ ). On a first step, for each linear operator 𝑢 ∈ L(𝑋∗ , 𝑌∗ ), we define, inspired by [12, Lemma 1],

sup Φ (𝜏𝑢∗∗ 𝐽𝑌 𝑦 (𝑓)) (𝑦) ‖𝑦‖ 0 such that Φ(𝑔)𝐵𝑌 (0,𝑅) ≤ 𝑔𝐵𝑋 (0,𝑆) ,

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= Φ (𝜏𝑢∗∗ 𝐽𝑌 𝑦 ∘ 𝜏V∗∗ 𝐽𝑌 𝑦 (𝑓)) (𝑦)

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∗ 𝑢+V (𝑓) (𝑦) . = Φ (𝜏(𝑢 ∗ +V∗ )𝐽𝑌 𝑦 (𝑓)) (𝑦) = Φ

Therefore, for 𝛿 = 𝜀 − ‖𝑢‖, we have 𝑉Ψ,𝛿 ⊂ 𝑉Φ,𝜀 and {𝑉Φ,𝜀 }𝜀>0 is a neighborhood basis at Φ. Also, for Φ ≠ Ψ ∈ M𝑏 (𝑋, 𝑌), we have that if 𝜋(Φ) = 𝜋(Ψ), then 𝑉Φ,𝜀 ∩ 𝑉Ψ,𝛿 = 0, for all 𝜀, 𝛿 > 0 and if 𝜋(Φ) ≠ 𝜋(Ψ), then 𝑉Φ,𝜀 ∩ 𝑉Ψ,𝜀 = 0 for 𝜀 = ‖𝜋(Φ) − 𝜋(Ψ)‖/2. This gives that the topology generated by {𝑉Φ,𝜀 }𝜀>0 is Hausdorff.

4

Abstract and Applied Analysis

Let us note that for each Φ in M𝑏 (𝑋, 𝑌) the subset 𝑉Φ = {Φ𝑢 : 𝑢 ∈ L(𝑋∗ , 𝑌∗ )} is the connected component containing Φ. Summing all this up we have proved the following result. Proposition 1. If 𝑋 is a symmetrically regular Banach space and 𝑌 is any Banach space, (M𝑏 (𝑋, 𝑌), 𝜋) is a Riemann domain over L(𝑋∗ , 𝑌∗ ) and each connected component of (M𝑏 (𝑋, 𝑌), 𝜋) is homeomorphic to L(𝑋∗ , 𝑌∗ ). Our aim now is to show that each function 𝑓 ∈ H𝑏 (𝑋) can be extended, in some sense, to a function on M𝑏 (𝑋, 𝑌) of bounded type. We do it with the following sort of Gelfand transform: 𝑓̂ : M𝑏 (𝑋, 𝑌) → H𝑏 (𝑌) , Φ → Φ (𝑓) ,

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and showing that this, when restricted to each connected component, is holomorphic of bounded type. To do that we need the following lemma. Lemma 2. If 𝑋 and 𝑌 are complex Banach spaces and 𝐺 is an element of H𝑏 (𝑋 × 𝑌), then the mapping 𝑔 defined by 𝑔(𝑥)(𝑦) = 𝐺(𝑥, 𝑦) for (𝑥, 𝑦) ∈ 𝑋×𝑌 belongs to H𝑏 (𝑋, H𝑏 (𝑌)). Conversely, given 𝑔 in H𝑏 (𝑋, H𝑏 (𝑌)) the mapping 𝐺(𝑥, 𝑦) = 𝑔(𝑥)(𝑦) belongs to H𝑏 (𝑋 × 𝑌). Proof. Let 𝐺 be in H𝑏 (𝑋 × 𝑌), and let ∑∞ 𝑚=0 𝑃𝑚 𝑓 be the Taylor series expansion of 𝐺 at (0, 0). We have ∞

𝑚=0

𝑚

∨ 𝑚 (𝑚−𝑘) = ∑ ∑ ( ) 𝑃𝑚 𝑓 ((𝑥, 0)(𝑘) , (0, 𝑦) ). 𝑘 𝑚=0

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𝑘=0

Note that if we take 𝑅, 𝑆 > 0 by the polarization formula (4) and Cauchy inequalities (6), we have ∞ 𝑚 ∨ 𝑚 (𝑚−𝑘) ) ∑ ∑ ( ) sup sup 𝑃𝑚 𝑓 ((𝑥, 0)(𝑘) , (0, 𝑦) 𝑘 ‖𝑥‖≤𝑅 𝑦 ≤𝑆 𝑚=0 𝑘=0 ‖ ‖ ∞ 𝑚 𝑚𝑚 𝑚 𝑘 𝑚−𝑘 ≤ ∑ ∑ ( ) sup sup 𝑃𝑚 𝑓 ‖𝑥‖ 𝑦 𝑘 𝑚! ‖𝑥‖≤𝑅 ‖𝑦‖≤𝑆 𝑚=0 𝑘=0 ∞

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≤ ∑ (𝑒 (𝑅 + 𝑆))𝑚 𝑚=0

× =2

1 𝑚 𝐺 (𝑥, 𝑦) + 𝑆)) (2𝑒 (𝑅 ‖𝑥‖≤2𝑒(𝑅+𝑆) ‖𝑦‖≤2𝑒(𝑅+𝑆)

sup

sup

sup

sup

‖𝑥‖≤2𝑒(𝑅+𝑆) ‖𝑦‖≤2𝑒(𝑅+𝑆)

𝐺 (𝑥, 𝑦) < ∞.

By the properties of convergent double series of nonnegative numbers we obtain that, for each fixed 𝑘, the series ∨

sup 𝐺 (𝑥, 𝑦) = sup sup 𝑔 (𝑥) (𝑦) < ∞, ‖𝑥‖≤𝑅,‖𝑦‖≤𝑆 ‖𝑦‖≤𝑆 ‖𝑥‖≤𝑅

(𝑘) (𝑚−𝑘) 𝑚 𝑄𝑘 (𝑥)(𝑦) := ∑∞ ) converges 𝑚=𝑘 ( 𝑘 ) 𝑃𝑚 𝑓 ((𝑥, 0) , (0, 𝑦)

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and 𝐺 is bounded on the bounded subsets of 𝑋 × 𝑌. Proposition 3. Let 𝑋 be a symmetrically regular Banach space and let 𝑌 be any Banach space. Given a function 𝑓 ∈ H𝑏 (𝑋) consider its extension 𝑓̂ : M𝑏 (𝑋, 𝑌) → H𝑏 (𝑌) defined in (24). We have that 𝑓̂ is a holomorphic function of bounded type. That is, 𝑓̂ ∘ (𝜋|𝑉Φ )−1 ∈ H𝑏 (L(𝑋∗ , 𝑌∗ ), H𝑏 (𝑌)) for every Φ. Proof. The point is to prove that the function L (𝑋∗ , 𝑌∗ ) → H𝑏 (𝑌) , 𝑢 → Φ𝑢 (𝑓)

𝐺 (𝑥, 𝑦) = ∑ 𝑃𝑚 𝑓 (𝑥, 𝑦) ∞

absolutely and uniformly on any product of balls in 𝑋 and 𝑌 with finite radii. Hence 𝑄𝑘 : 𝑋 → H𝑏 (𝑌) is a continuous 𝑘-homogeneous polynomial and actually 𝑔 = ∑∞ 𝑘=0 𝑄𝑘 is an entire function from 𝑋 to H𝑏 (𝑌) that, by above inequalities, is of bounded type. Conversely, consider 𝑔 in H𝑏 (𝑋, H𝑏 (𝑌)) and define 𝐺 : 𝑋 × 𝑌 → C by 𝐺(𝑥, 𝑦) = 𝑔(𝑥)(𝑦). By definition, for each 𝑥 ∈ 𝑋, 𝐺(𝑥, −) belongs to H𝑏 (𝑌). If we fix now 𝑦 ∈ 𝑌, we have that 𝛿𝑦 is a continuous linear form on H𝑏 (𝑌) and 𝐺(𝑥, 𝑦) = 𝛿𝑦 (𝑔(𝑥)), implying that 𝐺(−, 𝑦) is the composition of holomorphic mappings. Thus 𝐺(−, 𝑦) is holomorphic for every 𝑦 ∈ 𝑌. By Hartogs’ theorem, 𝐺 ∈ H(𝑋 × 𝑌). Finally, for fixed 𝑅, 𝑆 > 0 we have that 𝑔(𝐵𝑋 (0, 𝑅)) is a bounded subset of H𝑏 (𝑌). Hence

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is holomorphic of bounded type. For that, we introduce an auxiliary mapping 𝑀 : L(𝑋∗ , 𝑌∗ ) × 𝑌∗∗ × 𝑌 → C, defined by 𝑀 (𝑢, 𝑧, 𝑦) = Φ [𝑥 → 𝑓 (𝑥 + 𝑢∗ (𝑧))] (𝑦) .

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As above, we only need to check that it is separately holomorphic to conclude that 𝑀 is holomorphic. First we fix 𝑢 ∈ L(𝑋∗ , 𝑌∗ ) and 𝑧 ∈ 𝑌∗∗ and denote 𝑀𝑢,𝑧 (𝑦) := 𝑀(𝑢, 𝑧, 𝑦). We have 𝑀𝑢,𝑧 = Φ(𝜏𝑢∗∗ (𝑧) (𝑓)) and this belongs to H𝑏 (𝑌). Now we fix 𝑧, 𝑦 and take 𝑀𝑧,𝑦 (𝑢) := 𝑀(𝑢, 𝑧, 𝑦) = 𝛿𝑦 (Φ(𝜏𝑢∗∗ (𝑧) (𝑓))). This mapping is holomorphic (of bounded type) since it is the composition of the linear mapping L(𝑋∗ , 𝑌∗ ) → 𝑋∗∗ defined by 𝑢 → 𝑢∗ (𝑧) with the holomorphic mapping of bounded type: 𝑋∗∗ → C, V → (𝛿𝑦 ∘ Φ) (𝜏V∗ (𝑓)) .

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Finally we fix 𝑢, 𝑦 and denote 𝑀𝑢,𝑦 (𝑧) := 𝑀(𝑢, 𝑧, 𝑦) = 𝛿𝑦 (Φ(𝜏𝑢∗∗ (𝑧) (𝑓))). Again, this is the composition of a linear mapping 𝑌∗∗ → 𝑋∗∗ defined by 𝑧 → 𝑢∗ (𝑧) with the same holomorphic mapping (30). We conclude that 𝑀 is holomorphic.

Abstract and Applied Analysis

5

Let now 𝑅, 𝑆, 𝑇 > 0. Given 𝑇 > 0 there exists 𝑈 > 0 such that ‖Φ(ℎ)‖𝐵𝑌 (0,𝑇) ≤ ‖ℎ‖𝐵𝑋 (0,𝑈) for every ℎ ∈ H𝑏 (𝑋). Hence sup

‖𝑢‖≤𝑅,‖𝑧‖≤𝑆,‖𝑦‖≤𝑇

The mapping 𝜁(Φ) belongs to H𝑏 (𝑌, 𝑋∗∗ ) (and hence 𝜁 is well defined). This follows from the fact that 𝑌 × 𝑋∗ → C,

𝑀 (𝑢, 𝑧, 𝑦)

= sup sup Φ (𝜏𝑢∗∗ (𝑧) (𝑓))𝐵 (0,𝑇) 𝑌 ‖𝑢‖≤𝑅 ‖𝑧‖≤𝑆 ≤ sup sup 𝜏𝑢∗∗ (𝑧) (𝑓)𝐵 (0,𝑈) 𝑋 ‖𝑢‖≤𝑅 ‖𝑧‖≤𝑆

(𝑦, 𝑥∗ ) → Φ (𝑥∗ ) (𝑦) (31)

≤ 𝑓𝐵𝑋 (0,𝑈+𝑅𝑆) < ∞, and 𝑀 is of bounded type. Since the mapping 𝑌 → 𝑌∗∗ × 𝑌 defined by 𝑦 → (𝐽𝑌 𝑦, 𝑦) is obviously holomorphic of bounded type we have that 𝐺 : L (𝑋∗ , 𝑌∗ ) × 𝑌 → C

(32)

defined by 𝐺(𝑢, 𝑦) = 𝑀(𝑢, 𝑦, 𝑦) is holomorphic of bounded type. Now a direct application of Lemma 2 gives that the mapping 𝑢 → Φ𝑢 (𝑓) belongs to H𝑏 (L(𝑋∗ , 𝑌∗ ), H𝑏 (𝑌)). The above proposition is related to the study of extension of functions of bounded type given in [13].

4. The Size of the Fibers of M𝑏 (𝑋,𝑌) We focus now on the sets of elements in M𝑏 (𝑋, 𝑌) that are projected on the same element 𝑢 of L(𝑋∗ , 𝑌∗ ). This is called the “fiber” of 𝑢 and is defined by F (𝑢) = {Φ ∈ M𝑏 (𝑋, 𝑌) : 𝜋 (Φ) = 𝑢} .

(33)

Our aim in this section is to find out how big these fibers can be. To begin with, each fixed 𝑢 ∈ L(𝑋∗ , 𝑌∗ ) defines a set 𝐴 𝑢 = {𝑔 ∈ H𝑏 (𝑌, 𝑋∗∗ ) : 𝑑𝑔 (0) = 𝑢∗ ∘ 𝐽𝑌 } .

𝑗 : H𝑏 (𝑌, 𝑋∗∗ ) → M𝑏 (𝑋, 𝑌) , 𝑔 → Φ𝑔 ,

(35)

which maps the set 𝐴 𝑢 into the fiber F(𝑢). Let us check that 𝑗 is injective. Given 𝑔1 , 𝑔2 ∈ H𝑏 (𝑌, 𝑋∗∗ ) \ {0}, 𝑔1 ≠ 𝑔2 , there exists 𝑦0 ∈ 𝑌 such that 𝑔1 (𝑦0 ) ≠ 𝑔2 (𝑦0 ) and since 𝑋∗ separates points of 𝑋∗∗ we can find 𝑥0∗ ∈ 𝑋∗ with 𝑔1 (𝑦0 )(𝑥0∗ ) ≠ 𝑔2 (𝑦0 )(𝑥0∗ ). Thus Φ𝑔1 (𝑥0∗ )(𝑦0 ) ≠ Φ𝑔2 (𝑥0∗ )(𝑦0 ) and this gives Φ𝑔1 ≠ Φ𝑔2 . There is also a projection 𝜁 : M𝑏 (𝑋, 𝑌) → H𝑏 (𝑌, 𝑋∗∗ ) , Φ → [𝑦 → (𝑥∗ → Φ (𝑥∗ ) (𝑦))] .

is holomorphic. Clearly, (𝜁 ∘ 𝑗)(𝑔) = 𝑔. Also, note that 𝜁(Φ) determines the values that takes Φ when restricted to 𝑋∗ . This means that when finite type polynomials are dense in H𝑏 (𝑋), 𝜁(Φ) determines Φ. So, the only homomorphisms in M𝑏 (𝑋, 𝑌) are the Φ𝑔 ’s and we have the following result, which is closely related to [13, Lemmas 4.5 and 4.6] and in [1, Theorem 21]. Proposition 4. Let 𝑋 and 𝑌 be Banach spaces. If finite type polynomials are dense in H𝑏 (𝑋) then for each Φ ∈ M𝑏 (𝑋, 𝑌) there exists 𝑔 ∈ H𝑏 (𝑌, 𝑋∗∗ ) such that Φ = Φ𝑔 . Also, F(𝑢) = {Φ𝑔 : 𝑔 ∈ 𝐴 𝑢 }. Let us note that the mapping 𝑗|𝐴 𝑢 : 𝐴 𝑢 → F(𝑢) is actually injective and, by Proposition 4, if finite type polynomials are dense, surjective. This means that even in the case when finite type polynomials are dense in H𝑏 (𝑋) (i.e., the space H𝑏 (𝑋) is rather small), the fibers are thick. Let us see now that if this is not the case (i.e., when there is a polynomial in 𝑋 that is not weakly continuous on bounded sets), this mapping is no longer surjective and we can find even more homomorphisms inside each fiber. Proposition 5. If 𝑋 is symmetrically regular and there exists a polynomial on 𝑋 not weakly continuous on bounded sets at a point 𝑥0 , then, for each 𝑢 ∈ L(𝑋∗ , 𝑌∗ ), there is Φ ∈ F(𝑢) such that Φ ≠ Φ𝑔 , for all 𝑔 ∈ 𝐴 𝑢 . Proof. It is enough to prove the result for 𝐹(0) because we can change fibers through the mapping 𝐹 (0) → 𝐹 (𝑢) ,

(34)

On the other hand every 𝑔 ∈ H𝑏 (𝑌, 𝑋∗∗ ) defines a composition homomorphism Φ𝑔 ∈ M𝑏 (𝑋, 𝑌) given by Φ𝑔 (𝑓) = 𝑓 ∘ 𝑔. This gives an inclusion

(36)

(37)

(38)

Φ → Φ𝑢 .

Also, if Φ𝑔 ∈ 𝐹(0), then (Φ𝑔 )𝑢 = Φℎ , with ℎ(𝑦) = 𝑔(𝑦) + 𝑢∗ 𝐽𝑌 𝑦. Let 𝑃 be a polynomial that is not weakly continuous on bounded sets at 𝑥0 and {𝑥𝛼 } a bounded net weakly convergent to 𝑥0 such that |𝑃(𝑥𝛼 ) − 𝑃(𝑥0 )| > 1, for all 𝛼. For an ultrafilter U containing the sets {𝛼 : 𝛼 ≥ 𝛼0 }, let Φ be given by Φ (𝑓) (𝑦) = lim𝑓 (𝑥𝛼 ) , U

∀𝑓 ∈ H𝑏 (𝑋) , 𝑦 ∈ 𝑌.

(39)

Then Φ is a homomorphism in M𝑏 (𝑋, 𝑌) (actually in 𝐹(0)), that is, not of composition type. Indeed, since Φ(𝑓) is a constant function on 𝑌 it follows that 𝑑(Φ(𝑓))(0) = 0 for every 𝑓 in H𝑏 (𝑋) and so Φ ∈ 𝐹(0). If Φ = Φ𝑔 for certain 𝑔 ∈ H𝑏 (𝑌, 𝑋∗∗ ), then we have 𝑥∗ (𝑥0 ) = lim 𝑥∗ (𝑥𝛼 ) = 𝑥∗ (𝑔 (𝑦)) , U

∀𝑥∗ ∈ 𝑋∗ , 𝑦 ∈ 𝑌. (40)

6

Abstract and Applied Analysis

This says that 𝑔(𝑦) = 𝐽𝑋 𝑥0 , for all 𝑦. Hence, lim 𝑃 (𝑥𝛼 ) = 𝑃 (𝑔 (𝑦)) = 𝑃 (𝐽𝑋 𝑥0 ) = 𝑃 (𝑥0 ) , U

(41)

which is a contradiction. Something more can be said when there is a polynomial that is not weakly continuous on bounded sets. In this case, we can insert in each fiber a big set of homomorphisms that are not of composition type. We do it in detail in the fiber of 0. First, note that if there is a polynomial not weakly continuous on bounded sets, then there is a homogeneous polynomial 𝑃 which is not weakly continuous on bounded sets at 0. So, the Aron-Berner extension of 𝑃 is not 𝑤∗ continuous at any point 𝑥∗∗ ∈ 𝑋∗∗ [14, Corollary 2, Proposition 3] and [15, Proposition 1] (see also [16, Proof of Proposition 2.6]). Thus, we fix, for each 𝑥∗∗ ∈ 𝑋∗∗ a bounded net {𝑥𝛼∗∗ } in 𝑋∗∗ which 𝑤∗ -converges to 𝑥∗∗ , but |𝑃(𝑥𝛼∗∗ ) − 𝑃(𝑥∗∗ )| > 1, for all 𝛼. For each 𝑥∗∗ ∈ 𝑋∗∗ we fix also an ultrafilter U containing the sets {𝛼 : 𝛼 ≥ 𝛼0 }. Consider the set 𝐴 = {𝑔 ∈ H𝑏 (𝑌, 𝑋∗∗ ) : 𝑔 (0) = 0, 𝑑𝑔 (0) ≡ 0}

(42)

and define the following mapping 𝐴 × 𝑋∗∗ → 𝐹 (0) ,

(43)

(𝑔, 𝑥∗∗ ) → Ψ𝑔,𝑥∗∗ , Ψ𝑔,𝑥∗∗ (𝑓) (𝑦) = lim𝑓 (𝑥𝛼∗∗ + 𝑔 (𝑦)) . U

(44)

This mapping is well defined because Ψ𝑔,𝑥∗∗ (𝑥∗ )(𝑦) = 𝑥∗ (𝑥𝛼∗∗ + 𝑔(𝑦)) = 𝑥∗∗ (𝑥∗ ) + 𝑥∗ (𝑔(𝑦)); then 𝜋(Ψ𝑔,𝑥∗∗ )(𝑥∗ ) = 𝑑(𝑥∗ ∘ 𝑔)(0) = 𝑥∗ ∘ 𝑑𝑔(0) ≡ 0. The mapping is also injective. Indeed, if Ψ𝑔,𝑥∗∗ = Ψℎ,𝑧∗∗ , then Ψ𝑔,𝑥∗∗ (𝑥∗ )(𝑦) = Ψℎ,𝑧∗∗ (𝑥∗ )(𝑦), for all 𝑥∗ ∈ 𝑋∗ and 𝑦 ∈ 𝑌. Then, 𝑥

∗

(𝑥 ) +

𝑥∗

(𝑔 (𝑦)) = 𝑧

∗∗

∗

𝑥∗

(ℎ (𝑦)) ,

∀𝑥∗ ∈ 𝑋∗ ,

𝑦 ∈ 𝑌.

(𝑥 ) +

(45)

Therefore, ℎ(𝑦) = 𝑔(𝑦) + 𝑥∗∗ − 𝑧∗∗ , for all 𝑦 and, evaluating at 0, we obtain that 𝑥∗∗ = 𝑧∗∗ and, hence, 𝑔 = ℎ. Note, also, that the homomorphisms Ψ𝑔,𝑥∗∗ are not of composition type. Indeed, if Ψ𝑔,𝑥∗∗ = Φℎ , for certain 𝑔 ∈ 𝐴, 𝑥∗∗ ∈ 𝑋∗∗ , and ℎ ∈ H𝑏 (𝑌, 𝑋∗∗ ), then, for all 𝑥∗ ∈ 𝑋∗ , 𝑥∗∗ (𝑥∗ ) + 𝑥∗ ∘ 𝑔 = Ψ𝑔,𝑥∗∗ (𝑥∗ ) = Φℎ (𝑥∗ ) = 𝑥∗ ∘ ℎ. (46) If this were the case we would have ℎ(𝑦) = 𝑥∗∗ + 𝑔(𝑦), for all 𝑦. Now, lim 𝑃 (𝑥𝛼∗∗ + 𝑔 (𝑦))

Proof. Let 𝐾 be a finite dimensional compact subset of 𝐸. By hypothesis, Ψ(𝐾) is a compact subset of 𝑋 and hence it is bounded. Thus, there exists 𝑀 > 0 such that sup {𝑥∗ ∘ Ψ (𝑦) : 𝑥∗ ∈ 𝐴, 𝑦 ∈ 𝐾}

= sup {Ψ (𝑦) : 𝑦 ∈ 𝐾} < 𝑀.

= Ψ𝑔,𝑥∗∗ (𝑃) (𝑦) = Φℎ (𝑃) (𝑦) = 𝑃 (𝑥

∗∗

+ 𝑔 (𝑦)) . (47)

(48)

Hence, the family of scalar valued holomorphic functions {𝑥∗ ∘ Ψ : 𝑥∗ ∈ 𝐴} is bounded on the finite dimensional compact subsets of 𝐸. But as 𝐸 is a Fr´echet space, then it is also Baire, and by [11] this family is locally bounded: given 𝑦0 ∈ 𝐸 there exists an open neighborhood 𝑉 of 𝑦0 such that

= sup {𝑥∗ ∘ Ψ (𝑦)) : 𝑥∗ ∈ 𝐴, 𝑦 ∈ 𝑉} < ∞.

(49)

We have obtained that the Gˆateaux holomorphic function Ψ is locally bounded. Then it is holomorphic by [4, Proposition 3.7]. The following proposition gives that Λ is holomorphic. The proof may seem at some points similar to that of Proposition 3, but the fact that in the target space we have now a nonmetrizable locally convex topology makes the whole situation much more delicate. We are going to consider now in L(H𝑏 (𝑋), H𝑏 (𝑌)) the topology 𝜏𝛽 defined by the following fundamental system of seminorms: 𝑞𝑅,B (𝑇) = sup {𝑇 (𝑓) (𝑦) : 𝑦 ∈ 𝑌, 𝑦 ≤ 𝑅, 𝑓 ∈ B} , (50) for 𝑇 ∈ L(H𝑏 (𝑋), H𝑏 (𝑌)), where 𝑅 > 0 and B is a bounded subset of H𝑏 (𝑋). Proposition 7. The mapping Λ : H𝑏 (𝑌, 𝑋∗∗ ) → L (H𝑏 (𝑋) , H𝑏 (𝑌)) ,

U

Evaluating at 0 leads to a contradiction.

Lemma 6. let Ψ : 𝐸 → 𝑋 be a Gˆateaux holomorphic mapping from a complex Fr´echet space 𝐸 to a Banach space 𝑋 such that 𝑥∗ ∘ Ψ is holomorphic for every 𝑥∗ ∈ 𝐴, where 𝐴 is a norming subset of the closed unit ball of 𝑋∗ (i.e., ‖𝑥‖ = sup𝑥∗ ∈𝐴{|𝑥∗ (𝑥)|} for every 𝑥 ∈ 𝑋). Then Ψ is holomorphic on 𝐸.

sup {Ψ (𝑦) : 𝑦 ∈ 𝑉}

where

∗∗

Let us finish this analysis by studying Λ: the composition of the inclusion 𝑗 defined in (35) with the inclusion of M𝑏 (𝑋, 𝑌) into L(H𝑏 (𝑋), H𝑏 (𝑌)). Then the mapping Λ turns out to be holomorphic (endowing L(H𝑏 (𝑋), H𝑏 (𝑌)) with an appropriate topology). We prepare the proof of this fact with a lemma that is a variant of a classical Dunford result; see also [17, Theorem 3].

𝑔 → Λ (𝑔) = Φ𝑔

(51)

is injective and holomorphic if we consider in L(H𝑏 (𝑋), H𝑏 (𝑌)) the 𝜏𝛽 -topology.

Abstract and Applied Analysis

7

Proof. Clearly Λ is well defined. Our first step is to prove that Λ : H𝑏 (𝑌, 𝑋∗∗ ) → L𝜏𝛽 (H𝑏 (𝑋), H𝑏 (𝑌)) is Gˆateaux holomorphic. Let 𝑔1 , 𝑔2 ∈ H𝑏 (𝑌, 𝑋∗∗ ), 𝑓 ∈ H𝑏 (𝑋), 𝑦 ∈ 𝑌, and 𝑡 ∈ C. Here we have Λ(𝑔1 + 𝑡𝑔2 )(𝑓)(𝑦) = 𝑓(𝑔1 (𝑦) + 𝑡𝑔2 (𝑦)). If ∑∞ 𝑚=0 𝑃𝑚 𝑓 is the Taylor series expansion of 𝑓 at 0 on 𝑋, then ∞

∗∗

∗∗

𝑓 (𝑥 ) = ∑ 𝑃𝑚 𝑓 (𝑥 ) ,

and its sum is again 𝑆/(𝑆 − 𝑒(1 + 𝑠0 )𝑀). On the other hand, by using the polarization formula (4), [9, Theorem 3], and Cauchy’s inequalities (6) we get ∨ (𝑚−𝑘) (𝑘) , 𝑔2 (𝑦) ) sup 𝑃𝑚 𝑓 (𝑔1 (𝑦) ‖𝑦‖≤𝑅 𝑚−𝑘 𝑘 ≤ sup 𝑒𝑚 𝑃𝑚 𝑓 𝑔1 (𝑦) 𝑔2 (𝑦) ‖𝑦‖≤𝑅

(52)

𝑚=0

for every 𝑥∗∗ ∈ 𝑋∗∗ and the convergence is absolute and uniform on the bounded subsets of 𝑋∗∗ [8, 9]. Thus

By applying now (55) we obtain ∨ 𝑚 (𝑚−𝑘) (𝑘) 𝑘 , 𝑔2 (𝑦) ) 𝑠0 ∑ ∑ ( ) sup 𝑃𝑚 𝑓 (𝑔1 (𝑦) 𝑘 𝑦 ≤𝑅 𝑘=0 𝑚=𝑘 ‖ ‖

∞

= ∑ 𝑃𝑚 𝑓 (𝑔1 (𝑦) + 𝑡𝑔2 (𝑦))

𝑆 ≤ . 𝑓 𝑆 − 𝑒 (1 + 𝑠0 ) 𝑀 𝐵𝑋 (0,𝑆)

𝑚=0

∨

𝑚

𝑚 (𝑚−𝑘) (𝑘) = ∑ ∑ ( ) 𝑃𝑚 𝑓 (𝑔1 (𝑦) , (𝑡𝑔2 (𝑦)) ) 𝑘 𝑚=0 𝑘=0

∞

𝑒𝑀 𝑚 ) 𝑓𝐵𝑋 (0,𝑆) . 𝑆

∞ ∞

Λ (𝑔1 + 𝑡𝑔2 ) (𝑓) (𝑦)

∞

≤(

(53)

∨

𝑚

Since the function 𝑦 ∈ 𝑌 → 𝑃𝑚 𝑓 (𝑔1 (𝑦)(𝑚−𝑘) , 𝑔2 (𝑦)(𝑘) ) is the composition of a continuous multilinear mapping and two holomorphic mappings of bounded type, it is holomorphic of bounded type on 𝑌. By using (57), we obtain that the series

𝑘=0

∞

∨

where our last step is simply formal. We are going to concentrate our effort now to show that this formal last equality holds in our setting. If we denote by 𝐵𝑌 (0, 𝑅) the closure of 𝐵𝑌 (0, 𝑅), then 𝑔𝑗 (𝐵𝑌 (0, 𝑅)) is a bounded subset of 𝑋∗∗ for 𝑗 = 1, 2. Thus there exists 𝑀 > 0 such that ‖𝑔𝑗 (𝑦)‖ ≤ 𝑀, for every 𝑦 in 𝐵𝑌 (0, 𝑅) and 𝑗 = 1, 2. We fix 𝑠0 > 1 and we take 𝑆 > 0 such that (1 + 𝑠0 )𝑒𝑀 < 𝑆. We have ∞ 𝑚 𝑚 𝑀 𝑚−𝑘 𝑠 𝑀 𝑘 ∑ ∑ 𝑒𝑚 ( ) ( ) ( 0 ) 𝑘 𝑆 𝑆 𝑚=0

=

𝜏𝑏 -converges in H𝑏 (𝑌); hence 𝑇𝑘 (𝑓) belongs to H𝑏 (𝑌) for every 𝑓 in H𝑏 (𝑋). Actually, if we consider 𝑇𝑘 : H𝑏 (𝑋) → H𝑏 (𝑌), this is a linear operator. By (57), it is also continuous, since given 𝑅 > 0 there exists 𝑆 > 0 such that 𝑆 sup 𝑇𝑘 (𝑓) (𝑦) ≤ 𝑓𝐵𝑋 (0,𝑆) , 𝑆 − 𝑒 (1 + 𝑠 ) 𝑀 0 ‖𝑦‖≤𝑅

(59)

for every 𝑓 ∈ H𝑏 (𝑋). Then 𝑞 (𝑇𝑘 ) = sup sup 𝑇𝑘 (𝑓) (𝑦) ‖𝑦‖≤𝑅 𝑓∈B ∞

𝑚

𝑚 𝑒 (1 + 𝑀) ≤ ∑ ( )( ) sup 𝑓𝐵𝑋 (0,𝑆) . 𝑘 𝑆 𝑓∈B

𝑘=0

𝑚=0

(58)

𝑚=𝑘

𝑘=0𝑚=𝑘

∞

∨

𝑇𝑘 (𝑓) := ∑ 𝑃𝑚 𝑓 (𝑔1 (⋅)(𝑚−𝑘) , 𝑔2 (⋅)(𝑘) )

𝑚 (𝑚−𝑘) (𝑘) = ∑ ∑ ( ) 𝑃𝑚 𝑓 (𝑔1 (𝑦) , 𝑔2 (𝑦) ) 𝑡𝑘 , 𝑘

= ∑(

(57)

∨

𝑚 (𝑚−𝑘) (𝑘) = ∑ ∑ ( ) 𝑡𝑘 𝑃𝑚 𝑓 (𝑔1 (𝑦) , 𝑔2 (𝑦) ) 𝑘 𝑚=0 ∞ ∞

(56)

(60)

𝑚=𝑘

𝑚

𝑒 (1 + 𝑠0 ) 𝑀 ) 𝑆

(54)

We have, for 𝑡 ∈ C with |𝑡| ≤ 𝑠0 , again by (57), ∞

𝑆 . 𝑆 − 𝑒 (1 + 𝑠0 ) 𝑀

sup ∑ 𝑞 (𝑇𝑘 ) |𝑡|𝑘

|𝑡|≤𝑠0 𝑘=0

∞

Hence, by the properties of summability of double series of nonnegative numbers, the double series below is convergent in R:

= ∑ 𝑞 (𝑇𝑘 ) 𝑠0𝑘 ≤ 𝑘=0

𝑆 𝑆 − 𝑒 (1 + 𝑠0 ) 𝑀

(61)

× sup 𝑓𝐵𝑋 (0,𝑆) . 𝑓∈B

∞ ∞

𝑚 𝑀 ∑ ∑ 𝑒𝑚 ( ) ( ) 𝑘 𝑆

𝑘=0 𝑚=𝑘

𝑚−𝑘

(

𝑘

𝑠0 𝑀 ) < ∞, 𝑆

(55)

𝑘 As a consequence the series ∑∞ 𝑘=0 𝑇𝑘 𝑡 defined on C with values in L(H𝑏 (𝑋), H𝑏 (𝑌)), 𝜏𝛽 -converges uniformly on the

8

Abstract and Applied Analysis

compacts of C. Hence it is an entire function. Since we have proved that all series involved converge absolutely, we can apply the reordering of absolutely convergent double series to conclude that the last formal equality of (53) actually holds and then ∞

Λ (𝑔1 + 𝑡𝑔2 ) (𝑓) (𝑦) = ∑ 𝑇𝑘 𝑡𝑘

(62)

𝑘=0

is an entire function on C for every 𝑔1 , 𝑔2 . This gives that Λ is Gˆateaux holomorphic. We fix now 𝑞 = 𝑞𝑅,B , a continuous seminorm of the fundamental system defined in (50) and denote by 𝑍𝑞 the completion of the normed space (L(H𝑏 (𝑋), H𝑏 (𝑌))/ Ker 𝑞, 𝑞̂). Given 𝑦 ∈ 𝑌 and 𝑓 ∈ H𝑏 (𝑋) we define the continuous linear functional 𝛿𝑓,𝑦 : L (H𝑏 (𝑋) , H𝑏 (𝑌)) → C

(63)

by 𝛿𝑓,𝑦 (𝑇) = 𝑇(𝑓)(𝑦). Clearly the quotient mapping 𝛿̂𝑓,𝑦 : ̂ = 𝑇(𝑓)(𝑦), belongs to 𝑍∗ . On 𝑍𝑞 → C, defined by 𝛿̂𝑓,𝑦 (𝑇) 𝑞 the other hand the set {𝛿̂𝑓,𝑦 : ‖𝑦‖ ≤ 𝑅, 𝑓 ∈ B} is a norming subset of 𝑍𝑞 since 𝑞 (𝑇) = sup {𝛿𝑓,𝑦 (𝑇) : 𝑦 ∈ 𝑌, 𝑦 ≤ 𝑅, 𝑓 ∈ B} .

(64)

̂ : H𝑏 (𝑌, 𝑋∗∗ ) → 𝑍𝑞 that remains We can consider Λ ̂ : H𝑏 (𝑌, Gˆateaux holomorphic. Thus 𝛿𝑓,𝑦 ∘ Λ = 𝛿̂𝑓,𝑦 ∘ Λ ∗∗ 𝑋 ) → C is Gˆateaux holomorphic for every 𝑦 ∈ 𝑌 and every 𝑓 ∈ H𝑏 (𝑋). If we show that it is continuous, then it will be holomorphic and, by Lemma 6, we will get that ̂ : H𝑏 (𝑌, 𝑋∗∗ ) → 𝑍𝑞 is holomorphic for every seminorm Λ 𝑞. Since L𝜏𝛽 (H𝑏 (𝑋), H𝑏 (𝑌)) is a complete space, we can conclude that Λ : H𝑏 (𝑌, 𝑋∗∗ ) → L𝜏𝛽 (H𝑏 (𝑋), H𝑏 (𝑌)) is holomorphic. Let 𝑔 ∈ H𝑏 (𝑌, 𝑋∗∗ ). Now for fixed 𝑓 ∈ H𝑏 (𝑋) and 𝑦0 ∈ 𝑌, we consider 𝑅 > ‖𝑦0 ‖ and we choose 𝑆 > 0 such that 𝑔 (𝐵𝑌 (0, 𝑅)) ⊂ 𝐵𝑋∗∗ (0, 𝑆) .

(65)

As 𝑓 is uniformly continuous on bounded subsets of 𝑋∗∗ , for a given 𝜀 > 0, there exists 0 < 𝛿 < 1 such that if 𝑧1 , 𝑧2 ∈ 𝐵𝑋∗∗ (0, 𝑆 + 1) with ‖𝑧1 − 𝑧2 ‖ < 𝛿, then 𝑓 (𝑧1 ) − 𝑓 (𝑧2 ) < 𝜀.

(66)

Let ℎ ∈ H𝑏 (𝑌, 𝑋∗∗ ) with sup‖𝑦‖ 0, where Φ𝑢 (𝑓) (𝑦) = Φ (𝜏𝑢∗∗ 𝐽𝑌 𝑦 (𝑓)) (𝑦) = Φ [𝑥 → 𝑓 (𝐽𝑋 𝑥 + 𝑢∗ 𝐽𝑌 𝑦)] (𝑦) ,

(70)

for all 𝑓 ∈ H𝑏 (𝑋) and 𝑦 ∈ 𝐵𝑌 . The fact that Φ𝑢 (𝑓) is in H∞ (𝐵𝑌 ) follows from a similar argument to that in (18) taking 𝑅 = 1. Now, as in (24), we can define a Gelfand transform of 𝑓 ∈ H𝑏 (𝑋) by 𝑓̂ : M𝑏,∞ (𝑋, 𝐵𝑌 ) → H∞ (𝐵𝑌 ) , Φ → 𝑓̂ (Φ) = Φ (𝑓) ,

(71)

and we can see that this is a holomorphic extension of 𝑓 to M𝑏,∞ (𝑋, 𝐵𝑌 ). Proposition 10. Let 𝑋 be a symmetrically regular Banach space and let 𝑌 be any Banach space. Given a function 𝑓 ∈ H𝑏 (𝑋) consider its extension 𝑓̂ defined in (71). Then the restriction of 𝑓̂ to each connected component of M𝑏,∞ (𝑋, 𝐵𝑌 ) is a holomorphic function of bounded type.

Abstract and Applied Analysis

9 ∞

‖Φ (ℎ)‖ = sup |Φ (ℎ) (𝑧)| ≤ sup |ℎ (𝑥)| , ‖𝑧‖ 0 there exists 𝛿 > 0 such that if 𝑧1 , 𝑧2 are in 𝐵𝑋∗∗ (0, 𝑅 + 𝑀) with ‖𝑧1 − 𝑧2 ‖ < 𝛿, then |𝑓(𝑧1 ) − 𝑓(𝑧2 )| < 𝜀. Consider now 𝑢1 , 𝑢2 in L(𝑋∗ , 𝑌∗ ) with ‖𝑢𝑗 ‖ < 𝑀 for 𝑗 = 1, 2 and ‖𝑢1 − 𝑢2 ‖ < 𝛿. We have 𝑇 (𝑢1 ) − 𝑇 (𝑢2 ) = sup Φ𝑢1 (𝑓) (𝑦) − Φ𝑢2 (𝑓) (𝑦) ‖𝑦‖

Research Article Homomorphisms between Algebras of Holomorphic Functions Verónica Dimant,1 Domingo García,2 Manuel Maestre,2 and Pablo Sevilla-Peris3 1

Departamento de Matem´atica, Universidad de San Andr´es, Vito Dumas 284, Victoria, B1644BID Buenos Aires, Argentina Departamento de An´alisis Matem´atico, Universidad de Valencia, Doctor Moliner 50, Burjasot, 46100 Valencia, Spain 3 Instituto Universitario de Matem´atica Pura y Aplicada, Universitat Polit`ecnica de Val`encia, 46022 Valencia, Spain 2

Correspondence should be addressed to Pablo Sevilla-Peris; [email protected] Received 26 December 2013; Accepted 13 March 2014; Published 12 May 2014 Academic Editor: Alfredo Peris Copyright © 2014 Ver´onica Dimant et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For two complex Banach spaces 𝑋 and 𝑌, in this paper, we study the generalized spectrum M𝑏 (𝑋, 𝑌) of all nonzero algebra homomorphisms from H𝑏 (𝑋), the algebra of all bounded type entire functions on 𝑋, into H𝑏 (𝑌). We endow M𝑏 (𝑋, 𝑌) with a structure of Riemann domain over L(𝑋∗ , 𝑌∗ ) whenever 𝑋 is symmetrically regular. The size of the fibers is also studied. Following the philosophy of (Aron et al., 1991), this is a step to study the set M𝑏,∞ (𝑋, 𝐵𝑌 ) of all nonzero algebra homomorphisms from H𝑏 (𝑋) into H∞ (𝐵𝑌 ) of bounded holomorphic functions on the open unit ball of 𝑌 and M∞ (𝐵𝑋 , 𝐵𝑌 ) of all nonzero algebra homomorphisms from H∞ (𝐵𝑋 ) into H∞ (𝐵𝑌 ).

1. Introduction The study of homomorphisms between topological algebras is one of the basic issues in this theory. Two are the main topological algebras that we come across when we deal with holomorphic functions on infinite dimensional spaces (see Section 2 for precise definitions): H𝑏 (𝑋), the holomorphic functions of bounded type (which is a Fr´echet algebra), and H∞ (𝐵𝑋 ), the bounded holomorphic functions on the open unit ball (which is a Banach algebra). Here, as a first step in the study of the set of homomorphisms between H∞ (𝐵𝑋 ) spaces, we mainly focus on algebras of holomorphic functions of bounded type and homomorphisms between them; 𝐿 : H𝑏 (𝑋) → H𝑏 (𝑌) (i.e., continuous, linear, and multiplicative mappings). These were already considered in [1]. There the focus was to study the homomorphisms as “individuals,” seeking properties of single ones. We have here a different interest: we treat them as a whole, considering the set

M𝑏 (𝑋, 𝑌) = M (H𝑏 (𝑋) , H𝑏 (𝑌)) = {Φ : H𝑏 (𝑋) → H𝑏 (𝑌) algebra homomorphisms} \ {0} .

(1)

We will call this set the generalized spectrum or simply the spectrum. Our main aim is to study M𝑏 (𝑋, 𝑌) and to define on it a topological and a differential structure. This problem has the same flavor as considering M𝑏 (𝑋), the spectrum of the algebra H𝑏 (𝑋) (i.e., the set of nonzero continuous, linear, and multiplicative Φ : H𝑏 (𝑋) → C). This was studied in [2, 3], where a structure of Riemannian manifold over the bidual 𝑋∗∗ was defined on it (see also [4, Section 3.6] for a very neat and nice presentation and [5–7] for similar results). Our approach is very much indebted to that in [2] and we get up to some point analogous results, defining on M𝑏 (𝑋, 𝑌) a Riemann structure over L(𝑋∗ , 𝑌∗ ) (note that 𝑋∗∗ = L(𝑋∗ , C)). We will also be interested in the fibers of elements of L(𝑋∗ , 𝑌∗ ). The outline of the paper is the following. In Section 3, for two complex Banach spaces 𝑋 and 𝑌, we study the generalized spectrum M𝑏 (𝑋, 𝑌) of all nonzero algebra homomorphisms from H𝑏 (𝑋) to H𝑏 (𝑌). We endow it with a structure of Riemann domain over L(𝑋∗ , 𝑌∗ ) whenever 𝑋 is symmetrically regular. In Section 4, we focus on the sets (fibers) of elements in M𝑏 (𝑋, 𝑌) that are projected on the same element 𝑢 of L(𝑋∗ , 𝑌∗ ). The size of these fibers is studied and we prove that they are big by showing that they contain big sets. Following the philosophy of [2], all about M𝑏 (𝑋, 𝑌) is a step to study in Section 5 the spectrum

2

Abstract and Applied Analysis

M𝑏,∞ (𝑋, 𝐵𝑌 ) of all nonzero algebra homomorphisms from H𝑏 (𝑋) to H∞ (𝐵𝑌 ) of bounded holomorphic functions on the open unit ball of 𝑌. Finally, in Section 6, we deal with the generalized spectrum M∞ (𝐵𝑋 , 𝐵𝑌 ) of all nonzero algebra homomorphisms from H∞ (𝐵𝑋 ) to H∞ (𝐵𝑌 ).

2. Definitions and Preliminaries Unless otherwise stated capital letters such as 𝑋, 𝑌, . . . will denote complex Banach spaces. The dual will be denoted by 𝑋∗ and the open ball of center 𝑥0 and radius 𝑟 > 0 by 𝐵𝑋 (𝑥0 , 𝑟). If 𝑥0 = 0 and 𝑟 = 1 we just write 𝐵𝑋 . The space of continuous, linear operators from 𝑋 to 𝑌 will be denoted by L(𝑋, 𝑌); this is a Banach space with the norm ‖𝑢‖ = sup𝑥∈𝐵𝑋 ‖𝑢(𝑥)‖. The adjoint operator of 𝑢 ∈ L(𝑋, 𝑌) will be denoted by 𝑢∗ ∈ L(𝑌∗ , 𝑋∗ ). We will denote the canonical inclusion of a space into its bidual by 𝐽𝑋 : 𝑋 → 𝑋∗∗ . Given two complex locally convex spaces 𝐸 and 𝐹, a mapping 𝑃 : 𝐸 → 𝐹 is a continuous 𝑚-homogeneous polynomial if there is a continuous 𝑚-linear mapping 𝐿 : 𝐸 × ⋅ ⋅ ⋅ × 𝐸 → 𝐹 such that 𝑃 (𝑥) = 𝐿 (𝑥, . . . , 𝑥)

for every 𝑥.

(2)

Throughout the paper every polynomial and multilinear mapping will be assumed to be continuous. An 𝑚-linear mapping 𝐿 is called symmetric if 𝐿(𝑥𝜎1 , . . . , 𝑥𝜎𝑚 ) = 𝐿(𝑥1 , . . . , 𝑥𝑚 ) for every permutation 𝜎 of {1, . . . , 𝑚}. Each 𝑚-homogeneous polynomial has a unique ∨

symmetric mapping (which we denote by 𝑃) satisfying (2). If 𝑃 : 𝑋 → 𝑌 is a continuous 𝑚-homogeneous polynomial between Banach spaces, the following expressions define norms for 𝑚-linear mappings and for 𝑚-homogeneous polynomials, respectively: ‖𝐿‖ = sup {𝐿 (𝑥1 , . . . , 𝑥𝑚 ) : 𝑥𝑗 ≤ 1, 𝑗 = 1, . . . , 𝑚} , ‖𝑃‖ = sup {‖𝑃 (𝑥)‖ : ‖𝑥‖ ≤ 1} .

(3)

(4)

Given an 𝑚-linear mapping 𝐿, the notation 𝐿(𝑥(𝑘) , 𝑦(𝑚−𝑘) ) will mean that 𝑥 is repeated 𝑘 times and 𝑦 is repeated 𝑚 − 𝑘 times. A mapping 𝑓 : 𝑈 ⊆ 𝐸 → 𝐹 is holomorphic on the open set 𝑈 if for every 𝑥0 in 𝑈, there exist (𝑃𝑚 𝑓(𝑥0 ))𝑚 , each one of them an 𝑚-homogeneous polynomial, such that the series ∞

𝑓 (𝑥) = ∑ 𝑃𝑚 𝑓 (𝑥0 ) (𝑥 − 𝑥0 )

A holomorphic function 𝑓 : 𝐸 → 𝐹 is of bounded type if it sends bounded subsets of 𝐸 to bounded sets of 𝐹. We denote by H𝑏 (𝐸, 𝐹) the space of holomorphic functions of bounded type from 𝐸 to 𝐹. If 𝐹 = C we simply write H𝑏 (𝐸). Given 𝑈 ⊆ 𝐸 we write 𝑓𝑈 = sup 𝑓 (𝑥) . (7) 𝑥∈𝑈

With this notation for 𝑋 a Banach space H𝑏 (𝑋) is a Fr´echet algebra endowed with the topology 𝜏𝑏 of uniform convergence on the bounded sets, whose seminorms are (for 𝑅 > 0) 𝑞𝑅 (𝑓) = 𝑓𝐵𝑋 (0,𝑅) = sup {𝑓 (𝑥) : ‖𝑥‖ < 𝑅} . (8) Let 𝑀 be a differential manifold on a complex Banach space 𝑋 and 𝐹 a locally convex space. A mapping 𝑓 : 𝑀 → 𝐹 is said to be holomorphic (of bounded type) if 𝑓 ∘ 𝜑−1 : Ω → 𝐹 is holomorphic (of bounded type) for every chart (𝜑, Ω) of 𝑀. Given 𝑥 ∈ 𝑋, we write 𝛿𝑥 for the evaluation mapping at 𝑥; that is, 𝛿𝑥 (𝑓) = 𝑓(𝑥) for all holomorphic 𝑓. Let 𝐿 : 𝑋×𝑋 → C be a continuous bilinear form. Fix 𝑥 ∈ 𝑋 and for 𝑤 ∈ 𝑋∗∗ let (𝑦𝛽 ) be a net in 𝑋 weak-star convergent to 𝑤. Since 𝐿(𝑥, −) ∈ 𝑋∗ , then there exists lim𝛽 𝐿(𝑥, 𝑦𝛽 ) := 𝐿(𝑥, 𝑤). Now, fix 𝑤 ∈ 𝑋∗∗ and for 𝑧 ∈ 𝑋∗∗ let (𝑥𝛼 ) be a net in 𝑋 weak-star convergent to 𝑧. Since 𝐿(−, 𝑤) ∈ 𝑋∗ , then there exists ̃ (𝑧, 𝑤) := lim 𝐿 (𝑥𝛼 , 𝑤) = lim lim 𝐿 (𝑥𝛼 , 𝑦𝛽 ) . 𝐿 𝛼 𝛼 𝛽

The polarization formula gives [4, Corollary 1.8] ∨ 𝑚𝑚 ‖𝑃‖ ≤ 𝑃 ≤ ‖𝑃‖ . 𝑚!

known [4, page 152-153] that 𝑓 is holomorphic if and only if it is Gˆateaux holomorphic and continuous. In the case that 𝑓 : 𝑋 → 𝑌, 𝑋 and 𝑌 being complex Banach spaces, 𝑓 is holomorphic on 𝑋 if and only if it is Fr´echet differentiable on 𝑋 and in that case 𝑃1 (𝑥) = 𝑑𝑓(𝑥) for every 𝑥. Also, by the Cauchy inequalities [4, Proposition 3.2] we have, for every 𝑚, sup 𝑃𝑚 (𝑥0 ) (𝑥) ≤ sup 𝑓 (𝑥) . (6) 𝑥∈𝐵𝑋 (0,𝑅) 𝑥∈𝐵𝑋 (𝑥0 ,𝑅)

(5)

𝑚=0

converges uniformly in some neighborhood of 𝑥0 contained in 𝑈. This is called the “Taylor series expansion” of 𝑓 at 𝑥0 . If 𝐸 and 𝐹 are Fr´echet spaces, then 𝑓 : 𝐸 → 𝐹 is Gˆateaux holomorphic if for every 𝑥, 𝑦 ∈ 𝐸 the function C → 𝐹, 𝑡 → 𝑓(𝑥 + 𝑡𝑦) is holomorphic on some neighborhood of 0. It is

(9)

Following this idea Aron and Berner showed in [8] that every function 𝑓 ∈ H𝑏 (𝑋) admits an extension to the bidual, called the Aron-Berner extension, 𝑓 ∈ H𝑏 (𝑋∗∗ ). By [9, Theorem 3], for every 𝑚-homogeneous polynomial 𝑃, we have ‖𝑃‖ = ‖𝑃‖. A Banach space 𝑋 is symmetrically regular if for all continuous symmetric bilinear form 𝐿 : 𝑋 × 𝑋 → C it follows that ̃ (𝑧, 𝑤) = lim lim 𝐿 (𝑥𝛼 , 𝑦𝛽 ) . 𝐿 𝛼 𝛽

(10)

We refer the reader to [4, 10, 11] for the general background on the theory of holomorphic functions on infinite dimensional spaces. We are going to work with the set M𝑏 (𝑋, 𝑌) of nonzero algebra homomorphisms between spaces of holomorphic functions of bounded type defined in (1). Observe that an idempotent element 𝑔 in H𝑏 (𝑌) satisfies that 𝑔(𝑌) is a subset of {0, 1} and 𝑔(𝑌) is a connected set. Hence, either 𝑔 ≡ 0 or 𝑔 ≡ 1. So, for any Φ ∈ M𝑏 (𝑋, 𝑌), we should have Φ(1𝑋 ) = 1𝑌 .

Abstract and Applied Analysis

3

3. The Differential Structure of M𝑏 (𝑋,𝑌)

for all 𝑔 ∈ H𝑏 (𝑋). Hence

Our aim in this section is to endow M𝑏 (𝑋, 𝑌) with a structure of Riemann domain over L(𝑋∗ , 𝑌∗ ). On a first step, for each linear operator 𝑢 ∈ L(𝑋∗ , 𝑌∗ ), we define, inspired by [12, Lemma 1],

sup Φ (𝜏𝑢∗∗ 𝐽𝑌 𝑦 (𝑓)) (𝑦) ‖𝑦‖ 0 such that Φ(𝑔)𝐵𝑌 (0,𝑅) ≤ 𝑔𝐵𝑋 (0,𝑆) ,

(17)

= Φ (𝜏𝑢∗∗ 𝐽𝑌 𝑦 ∘ 𝜏V∗∗ 𝐽𝑌 𝑦 (𝑓)) (𝑦)

(23)

∗ 𝑢+V (𝑓) (𝑦) . = Φ (𝜏(𝑢 ∗ +V∗ )𝐽𝑌 𝑦 (𝑓)) (𝑦) = Φ

Therefore, for 𝛿 = 𝜀 − ‖𝑢‖, we have 𝑉Ψ,𝛿 ⊂ 𝑉Φ,𝜀 and {𝑉Φ,𝜀 }𝜀>0 is a neighborhood basis at Φ. Also, for Φ ≠ Ψ ∈ M𝑏 (𝑋, 𝑌), we have that if 𝜋(Φ) = 𝜋(Ψ), then 𝑉Φ,𝜀 ∩ 𝑉Ψ,𝛿 = 0, for all 𝜀, 𝛿 > 0 and if 𝜋(Φ) ≠ 𝜋(Ψ), then 𝑉Φ,𝜀 ∩ 𝑉Ψ,𝜀 = 0 for 𝜀 = ‖𝜋(Φ) − 𝜋(Ψ)‖/2. This gives that the topology generated by {𝑉Φ,𝜀 }𝜀>0 is Hausdorff.

4

Abstract and Applied Analysis

Let us note that for each Φ in M𝑏 (𝑋, 𝑌) the subset 𝑉Φ = {Φ𝑢 : 𝑢 ∈ L(𝑋∗ , 𝑌∗ )} is the connected component containing Φ. Summing all this up we have proved the following result. Proposition 1. If 𝑋 is a symmetrically regular Banach space and 𝑌 is any Banach space, (M𝑏 (𝑋, 𝑌), 𝜋) is a Riemann domain over L(𝑋∗ , 𝑌∗ ) and each connected component of (M𝑏 (𝑋, 𝑌), 𝜋) is homeomorphic to L(𝑋∗ , 𝑌∗ ). Our aim now is to show that each function 𝑓 ∈ H𝑏 (𝑋) can be extended, in some sense, to a function on M𝑏 (𝑋, 𝑌) of bounded type. We do it with the following sort of Gelfand transform: 𝑓̂ : M𝑏 (𝑋, 𝑌) → H𝑏 (𝑌) , Φ → Φ (𝑓) ,

(24)

and showing that this, when restricted to each connected component, is holomorphic of bounded type. To do that we need the following lemma. Lemma 2. If 𝑋 and 𝑌 are complex Banach spaces and 𝐺 is an element of H𝑏 (𝑋 × 𝑌), then the mapping 𝑔 defined by 𝑔(𝑥)(𝑦) = 𝐺(𝑥, 𝑦) for (𝑥, 𝑦) ∈ 𝑋×𝑌 belongs to H𝑏 (𝑋, H𝑏 (𝑌)). Conversely, given 𝑔 in H𝑏 (𝑋, H𝑏 (𝑌)) the mapping 𝐺(𝑥, 𝑦) = 𝑔(𝑥)(𝑦) belongs to H𝑏 (𝑋 × 𝑌). Proof. Let 𝐺 be in H𝑏 (𝑋 × 𝑌), and let ∑∞ 𝑚=0 𝑃𝑚 𝑓 be the Taylor series expansion of 𝐺 at (0, 0). We have ∞

𝑚=0

𝑚

∨ 𝑚 (𝑚−𝑘) = ∑ ∑ ( ) 𝑃𝑚 𝑓 ((𝑥, 0)(𝑘) , (0, 𝑦) ). 𝑘 𝑚=0

(25)

𝑘=0

Note that if we take 𝑅, 𝑆 > 0 by the polarization formula (4) and Cauchy inequalities (6), we have ∞ 𝑚 ∨ 𝑚 (𝑚−𝑘) ) ∑ ∑ ( ) sup sup 𝑃𝑚 𝑓 ((𝑥, 0)(𝑘) , (0, 𝑦) 𝑘 ‖𝑥‖≤𝑅 𝑦 ≤𝑆 𝑚=0 𝑘=0 ‖ ‖ ∞ 𝑚 𝑚𝑚 𝑚 𝑘 𝑚−𝑘 ≤ ∑ ∑ ( ) sup sup 𝑃𝑚 𝑓 ‖𝑥‖ 𝑦 𝑘 𝑚! ‖𝑥‖≤𝑅 ‖𝑦‖≤𝑆 𝑚=0 𝑘=0 ∞

(26)

≤ ∑ (𝑒 (𝑅 + 𝑆))𝑚 𝑚=0

× =2

1 𝑚 𝐺 (𝑥, 𝑦) + 𝑆)) (2𝑒 (𝑅 ‖𝑥‖≤2𝑒(𝑅+𝑆) ‖𝑦‖≤2𝑒(𝑅+𝑆)

sup

sup

sup

sup

‖𝑥‖≤2𝑒(𝑅+𝑆) ‖𝑦‖≤2𝑒(𝑅+𝑆)

𝐺 (𝑥, 𝑦) < ∞.

By the properties of convergent double series of nonnegative numbers we obtain that, for each fixed 𝑘, the series ∨

sup 𝐺 (𝑥, 𝑦) = sup sup 𝑔 (𝑥) (𝑦) < ∞, ‖𝑥‖≤𝑅,‖𝑦‖≤𝑆 ‖𝑦‖≤𝑆 ‖𝑥‖≤𝑅

(𝑘) (𝑚−𝑘) 𝑚 𝑄𝑘 (𝑥)(𝑦) := ∑∞ ) converges 𝑚=𝑘 ( 𝑘 ) 𝑃𝑚 𝑓 ((𝑥, 0) , (0, 𝑦)

(27)

and 𝐺 is bounded on the bounded subsets of 𝑋 × 𝑌. Proposition 3. Let 𝑋 be a symmetrically regular Banach space and let 𝑌 be any Banach space. Given a function 𝑓 ∈ H𝑏 (𝑋) consider its extension 𝑓̂ : M𝑏 (𝑋, 𝑌) → H𝑏 (𝑌) defined in (24). We have that 𝑓̂ is a holomorphic function of bounded type. That is, 𝑓̂ ∘ (𝜋|𝑉Φ )−1 ∈ H𝑏 (L(𝑋∗ , 𝑌∗ ), H𝑏 (𝑌)) for every Φ. Proof. The point is to prove that the function L (𝑋∗ , 𝑌∗ ) → H𝑏 (𝑌) , 𝑢 → Φ𝑢 (𝑓)

𝐺 (𝑥, 𝑦) = ∑ 𝑃𝑚 𝑓 (𝑥, 𝑦) ∞

absolutely and uniformly on any product of balls in 𝑋 and 𝑌 with finite radii. Hence 𝑄𝑘 : 𝑋 → H𝑏 (𝑌) is a continuous 𝑘-homogeneous polynomial and actually 𝑔 = ∑∞ 𝑘=0 𝑄𝑘 is an entire function from 𝑋 to H𝑏 (𝑌) that, by above inequalities, is of bounded type. Conversely, consider 𝑔 in H𝑏 (𝑋, H𝑏 (𝑌)) and define 𝐺 : 𝑋 × 𝑌 → C by 𝐺(𝑥, 𝑦) = 𝑔(𝑥)(𝑦). By definition, for each 𝑥 ∈ 𝑋, 𝐺(𝑥, −) belongs to H𝑏 (𝑌). If we fix now 𝑦 ∈ 𝑌, we have that 𝛿𝑦 is a continuous linear form on H𝑏 (𝑌) and 𝐺(𝑥, 𝑦) = 𝛿𝑦 (𝑔(𝑥)), implying that 𝐺(−, 𝑦) is the composition of holomorphic mappings. Thus 𝐺(−, 𝑦) is holomorphic for every 𝑦 ∈ 𝑌. By Hartogs’ theorem, 𝐺 ∈ H(𝑋 × 𝑌). Finally, for fixed 𝑅, 𝑆 > 0 we have that 𝑔(𝐵𝑋 (0, 𝑅)) is a bounded subset of H𝑏 (𝑌). Hence

(28)

is holomorphic of bounded type. For that, we introduce an auxiliary mapping 𝑀 : L(𝑋∗ , 𝑌∗ ) × 𝑌∗∗ × 𝑌 → C, defined by 𝑀 (𝑢, 𝑧, 𝑦) = Φ [𝑥 → 𝑓 (𝑥 + 𝑢∗ (𝑧))] (𝑦) .

(29)

As above, we only need to check that it is separately holomorphic to conclude that 𝑀 is holomorphic. First we fix 𝑢 ∈ L(𝑋∗ , 𝑌∗ ) and 𝑧 ∈ 𝑌∗∗ and denote 𝑀𝑢,𝑧 (𝑦) := 𝑀(𝑢, 𝑧, 𝑦). We have 𝑀𝑢,𝑧 = Φ(𝜏𝑢∗∗ (𝑧) (𝑓)) and this belongs to H𝑏 (𝑌). Now we fix 𝑧, 𝑦 and take 𝑀𝑧,𝑦 (𝑢) := 𝑀(𝑢, 𝑧, 𝑦) = 𝛿𝑦 (Φ(𝜏𝑢∗∗ (𝑧) (𝑓))). This mapping is holomorphic (of bounded type) since it is the composition of the linear mapping L(𝑋∗ , 𝑌∗ ) → 𝑋∗∗ defined by 𝑢 → 𝑢∗ (𝑧) with the holomorphic mapping of bounded type: 𝑋∗∗ → C, V → (𝛿𝑦 ∘ Φ) (𝜏V∗ (𝑓)) .

(30)

Finally we fix 𝑢, 𝑦 and denote 𝑀𝑢,𝑦 (𝑧) := 𝑀(𝑢, 𝑧, 𝑦) = 𝛿𝑦 (Φ(𝜏𝑢∗∗ (𝑧) (𝑓))). Again, this is the composition of a linear mapping 𝑌∗∗ → 𝑋∗∗ defined by 𝑧 → 𝑢∗ (𝑧) with the same holomorphic mapping (30). We conclude that 𝑀 is holomorphic.

Abstract and Applied Analysis

5

Let now 𝑅, 𝑆, 𝑇 > 0. Given 𝑇 > 0 there exists 𝑈 > 0 such that ‖Φ(ℎ)‖𝐵𝑌 (0,𝑇) ≤ ‖ℎ‖𝐵𝑋 (0,𝑈) for every ℎ ∈ H𝑏 (𝑋). Hence sup

‖𝑢‖≤𝑅,‖𝑧‖≤𝑆,‖𝑦‖≤𝑇

The mapping 𝜁(Φ) belongs to H𝑏 (𝑌, 𝑋∗∗ ) (and hence 𝜁 is well defined). This follows from the fact that 𝑌 × 𝑋∗ → C,

𝑀 (𝑢, 𝑧, 𝑦)

= sup sup Φ (𝜏𝑢∗∗ (𝑧) (𝑓))𝐵 (0,𝑇) 𝑌 ‖𝑢‖≤𝑅 ‖𝑧‖≤𝑆 ≤ sup sup 𝜏𝑢∗∗ (𝑧) (𝑓)𝐵 (0,𝑈) 𝑋 ‖𝑢‖≤𝑅 ‖𝑧‖≤𝑆

(𝑦, 𝑥∗ ) → Φ (𝑥∗ ) (𝑦) (31)

≤ 𝑓𝐵𝑋 (0,𝑈+𝑅𝑆) < ∞, and 𝑀 is of bounded type. Since the mapping 𝑌 → 𝑌∗∗ × 𝑌 defined by 𝑦 → (𝐽𝑌 𝑦, 𝑦) is obviously holomorphic of bounded type we have that 𝐺 : L (𝑋∗ , 𝑌∗ ) × 𝑌 → C

(32)

defined by 𝐺(𝑢, 𝑦) = 𝑀(𝑢, 𝑦, 𝑦) is holomorphic of bounded type. Now a direct application of Lemma 2 gives that the mapping 𝑢 → Φ𝑢 (𝑓) belongs to H𝑏 (L(𝑋∗ , 𝑌∗ ), H𝑏 (𝑌)). The above proposition is related to the study of extension of functions of bounded type given in [13].

4. The Size of the Fibers of M𝑏 (𝑋,𝑌) We focus now on the sets of elements in M𝑏 (𝑋, 𝑌) that are projected on the same element 𝑢 of L(𝑋∗ , 𝑌∗ ). This is called the “fiber” of 𝑢 and is defined by F (𝑢) = {Φ ∈ M𝑏 (𝑋, 𝑌) : 𝜋 (Φ) = 𝑢} .

(33)

Our aim in this section is to find out how big these fibers can be. To begin with, each fixed 𝑢 ∈ L(𝑋∗ , 𝑌∗ ) defines a set 𝐴 𝑢 = {𝑔 ∈ H𝑏 (𝑌, 𝑋∗∗ ) : 𝑑𝑔 (0) = 𝑢∗ ∘ 𝐽𝑌 } .

𝑗 : H𝑏 (𝑌, 𝑋∗∗ ) → M𝑏 (𝑋, 𝑌) , 𝑔 → Φ𝑔 ,

(35)

which maps the set 𝐴 𝑢 into the fiber F(𝑢). Let us check that 𝑗 is injective. Given 𝑔1 , 𝑔2 ∈ H𝑏 (𝑌, 𝑋∗∗ ) \ {0}, 𝑔1 ≠ 𝑔2 , there exists 𝑦0 ∈ 𝑌 such that 𝑔1 (𝑦0 ) ≠ 𝑔2 (𝑦0 ) and since 𝑋∗ separates points of 𝑋∗∗ we can find 𝑥0∗ ∈ 𝑋∗ with 𝑔1 (𝑦0 )(𝑥0∗ ) ≠ 𝑔2 (𝑦0 )(𝑥0∗ ). Thus Φ𝑔1 (𝑥0∗ )(𝑦0 ) ≠ Φ𝑔2 (𝑥0∗ )(𝑦0 ) and this gives Φ𝑔1 ≠ Φ𝑔2 . There is also a projection 𝜁 : M𝑏 (𝑋, 𝑌) → H𝑏 (𝑌, 𝑋∗∗ ) , Φ → [𝑦 → (𝑥∗ → Φ (𝑥∗ ) (𝑦))] .

is holomorphic. Clearly, (𝜁 ∘ 𝑗)(𝑔) = 𝑔. Also, note that 𝜁(Φ) determines the values that takes Φ when restricted to 𝑋∗ . This means that when finite type polynomials are dense in H𝑏 (𝑋), 𝜁(Φ) determines Φ. So, the only homomorphisms in M𝑏 (𝑋, 𝑌) are the Φ𝑔 ’s and we have the following result, which is closely related to [13, Lemmas 4.5 and 4.6] and in [1, Theorem 21]. Proposition 4. Let 𝑋 and 𝑌 be Banach spaces. If finite type polynomials are dense in H𝑏 (𝑋) then for each Φ ∈ M𝑏 (𝑋, 𝑌) there exists 𝑔 ∈ H𝑏 (𝑌, 𝑋∗∗ ) such that Φ = Φ𝑔 . Also, F(𝑢) = {Φ𝑔 : 𝑔 ∈ 𝐴 𝑢 }. Let us note that the mapping 𝑗|𝐴 𝑢 : 𝐴 𝑢 → F(𝑢) is actually injective and, by Proposition 4, if finite type polynomials are dense, surjective. This means that even in the case when finite type polynomials are dense in H𝑏 (𝑋) (i.e., the space H𝑏 (𝑋) is rather small), the fibers are thick. Let us see now that if this is not the case (i.e., when there is a polynomial in 𝑋 that is not weakly continuous on bounded sets), this mapping is no longer surjective and we can find even more homomorphisms inside each fiber. Proposition 5. If 𝑋 is symmetrically regular and there exists a polynomial on 𝑋 not weakly continuous on bounded sets at a point 𝑥0 , then, for each 𝑢 ∈ L(𝑋∗ , 𝑌∗ ), there is Φ ∈ F(𝑢) such that Φ ≠ Φ𝑔 , for all 𝑔 ∈ 𝐴 𝑢 . Proof. It is enough to prove the result for 𝐹(0) because we can change fibers through the mapping 𝐹 (0) → 𝐹 (𝑢) ,

(34)

On the other hand every 𝑔 ∈ H𝑏 (𝑌, 𝑋∗∗ ) defines a composition homomorphism Φ𝑔 ∈ M𝑏 (𝑋, 𝑌) given by Φ𝑔 (𝑓) = 𝑓 ∘ 𝑔. This gives an inclusion

(36)

(37)

(38)

Φ → Φ𝑢 .

Also, if Φ𝑔 ∈ 𝐹(0), then (Φ𝑔 )𝑢 = Φℎ , with ℎ(𝑦) = 𝑔(𝑦) + 𝑢∗ 𝐽𝑌 𝑦. Let 𝑃 be a polynomial that is not weakly continuous on bounded sets at 𝑥0 and {𝑥𝛼 } a bounded net weakly convergent to 𝑥0 such that |𝑃(𝑥𝛼 ) − 𝑃(𝑥0 )| > 1, for all 𝛼. For an ultrafilter U containing the sets {𝛼 : 𝛼 ≥ 𝛼0 }, let Φ be given by Φ (𝑓) (𝑦) = lim𝑓 (𝑥𝛼 ) , U

∀𝑓 ∈ H𝑏 (𝑋) , 𝑦 ∈ 𝑌.

(39)

Then Φ is a homomorphism in M𝑏 (𝑋, 𝑌) (actually in 𝐹(0)), that is, not of composition type. Indeed, since Φ(𝑓) is a constant function on 𝑌 it follows that 𝑑(Φ(𝑓))(0) = 0 for every 𝑓 in H𝑏 (𝑋) and so Φ ∈ 𝐹(0). If Φ = Φ𝑔 for certain 𝑔 ∈ H𝑏 (𝑌, 𝑋∗∗ ), then we have 𝑥∗ (𝑥0 ) = lim 𝑥∗ (𝑥𝛼 ) = 𝑥∗ (𝑔 (𝑦)) , U

∀𝑥∗ ∈ 𝑋∗ , 𝑦 ∈ 𝑌. (40)

6

Abstract and Applied Analysis

This says that 𝑔(𝑦) = 𝐽𝑋 𝑥0 , for all 𝑦. Hence, lim 𝑃 (𝑥𝛼 ) = 𝑃 (𝑔 (𝑦)) = 𝑃 (𝐽𝑋 𝑥0 ) = 𝑃 (𝑥0 ) , U

(41)

which is a contradiction. Something more can be said when there is a polynomial that is not weakly continuous on bounded sets. In this case, we can insert in each fiber a big set of homomorphisms that are not of composition type. We do it in detail in the fiber of 0. First, note that if there is a polynomial not weakly continuous on bounded sets, then there is a homogeneous polynomial 𝑃 which is not weakly continuous on bounded sets at 0. So, the Aron-Berner extension of 𝑃 is not 𝑤∗ continuous at any point 𝑥∗∗ ∈ 𝑋∗∗ [14, Corollary 2, Proposition 3] and [15, Proposition 1] (see also [16, Proof of Proposition 2.6]). Thus, we fix, for each 𝑥∗∗ ∈ 𝑋∗∗ a bounded net {𝑥𝛼∗∗ } in 𝑋∗∗ which 𝑤∗ -converges to 𝑥∗∗ , but |𝑃(𝑥𝛼∗∗ ) − 𝑃(𝑥∗∗ )| > 1, for all 𝛼. For each 𝑥∗∗ ∈ 𝑋∗∗ we fix also an ultrafilter U containing the sets {𝛼 : 𝛼 ≥ 𝛼0 }. Consider the set 𝐴 = {𝑔 ∈ H𝑏 (𝑌, 𝑋∗∗ ) : 𝑔 (0) = 0, 𝑑𝑔 (0) ≡ 0}

(42)

and define the following mapping 𝐴 × 𝑋∗∗ → 𝐹 (0) ,

(43)

(𝑔, 𝑥∗∗ ) → Ψ𝑔,𝑥∗∗ , Ψ𝑔,𝑥∗∗ (𝑓) (𝑦) = lim𝑓 (𝑥𝛼∗∗ + 𝑔 (𝑦)) . U

(44)

This mapping is well defined because Ψ𝑔,𝑥∗∗ (𝑥∗ )(𝑦) = 𝑥∗ (𝑥𝛼∗∗ + 𝑔(𝑦)) = 𝑥∗∗ (𝑥∗ ) + 𝑥∗ (𝑔(𝑦)); then 𝜋(Ψ𝑔,𝑥∗∗ )(𝑥∗ ) = 𝑑(𝑥∗ ∘ 𝑔)(0) = 𝑥∗ ∘ 𝑑𝑔(0) ≡ 0. The mapping is also injective. Indeed, if Ψ𝑔,𝑥∗∗ = Ψℎ,𝑧∗∗ , then Ψ𝑔,𝑥∗∗ (𝑥∗ )(𝑦) = Ψℎ,𝑧∗∗ (𝑥∗ )(𝑦), for all 𝑥∗ ∈ 𝑋∗ and 𝑦 ∈ 𝑌. Then, 𝑥

∗

(𝑥 ) +

𝑥∗

(𝑔 (𝑦)) = 𝑧

∗∗

∗

𝑥∗

(ℎ (𝑦)) ,

∀𝑥∗ ∈ 𝑋∗ ,

𝑦 ∈ 𝑌.

(𝑥 ) +

(45)

Therefore, ℎ(𝑦) = 𝑔(𝑦) + 𝑥∗∗ − 𝑧∗∗ , for all 𝑦 and, evaluating at 0, we obtain that 𝑥∗∗ = 𝑧∗∗ and, hence, 𝑔 = ℎ. Note, also, that the homomorphisms Ψ𝑔,𝑥∗∗ are not of composition type. Indeed, if Ψ𝑔,𝑥∗∗ = Φℎ , for certain 𝑔 ∈ 𝐴, 𝑥∗∗ ∈ 𝑋∗∗ , and ℎ ∈ H𝑏 (𝑌, 𝑋∗∗ ), then, for all 𝑥∗ ∈ 𝑋∗ , 𝑥∗∗ (𝑥∗ ) + 𝑥∗ ∘ 𝑔 = Ψ𝑔,𝑥∗∗ (𝑥∗ ) = Φℎ (𝑥∗ ) = 𝑥∗ ∘ ℎ. (46) If this were the case we would have ℎ(𝑦) = 𝑥∗∗ + 𝑔(𝑦), for all 𝑦. Now, lim 𝑃 (𝑥𝛼∗∗ + 𝑔 (𝑦))

Proof. Let 𝐾 be a finite dimensional compact subset of 𝐸. By hypothesis, Ψ(𝐾) is a compact subset of 𝑋 and hence it is bounded. Thus, there exists 𝑀 > 0 such that sup {𝑥∗ ∘ Ψ (𝑦) : 𝑥∗ ∈ 𝐴, 𝑦 ∈ 𝐾}

= sup {Ψ (𝑦) : 𝑦 ∈ 𝐾} < 𝑀.

= Ψ𝑔,𝑥∗∗ (𝑃) (𝑦) = Φℎ (𝑃) (𝑦) = 𝑃 (𝑥

∗∗

+ 𝑔 (𝑦)) . (47)

(48)

Hence, the family of scalar valued holomorphic functions {𝑥∗ ∘ Ψ : 𝑥∗ ∈ 𝐴} is bounded on the finite dimensional compact subsets of 𝐸. But as 𝐸 is a Fr´echet space, then it is also Baire, and by [11] this family is locally bounded: given 𝑦0 ∈ 𝐸 there exists an open neighborhood 𝑉 of 𝑦0 such that

= sup {𝑥∗ ∘ Ψ (𝑦)) : 𝑥∗ ∈ 𝐴, 𝑦 ∈ 𝑉} < ∞.

(49)

We have obtained that the Gˆateaux holomorphic function Ψ is locally bounded. Then it is holomorphic by [4, Proposition 3.7]. The following proposition gives that Λ is holomorphic. The proof may seem at some points similar to that of Proposition 3, but the fact that in the target space we have now a nonmetrizable locally convex topology makes the whole situation much more delicate. We are going to consider now in L(H𝑏 (𝑋), H𝑏 (𝑌)) the topology 𝜏𝛽 defined by the following fundamental system of seminorms: 𝑞𝑅,B (𝑇) = sup {𝑇 (𝑓) (𝑦) : 𝑦 ∈ 𝑌, 𝑦 ≤ 𝑅, 𝑓 ∈ B} , (50) for 𝑇 ∈ L(H𝑏 (𝑋), H𝑏 (𝑌)), where 𝑅 > 0 and B is a bounded subset of H𝑏 (𝑋). Proposition 7. The mapping Λ : H𝑏 (𝑌, 𝑋∗∗ ) → L (H𝑏 (𝑋) , H𝑏 (𝑌)) ,

U

Evaluating at 0 leads to a contradiction.

Lemma 6. let Ψ : 𝐸 → 𝑋 be a Gˆateaux holomorphic mapping from a complex Fr´echet space 𝐸 to a Banach space 𝑋 such that 𝑥∗ ∘ Ψ is holomorphic for every 𝑥∗ ∈ 𝐴, where 𝐴 is a norming subset of the closed unit ball of 𝑋∗ (i.e., ‖𝑥‖ = sup𝑥∗ ∈𝐴{|𝑥∗ (𝑥)|} for every 𝑥 ∈ 𝑋). Then Ψ is holomorphic on 𝐸.

sup {Ψ (𝑦) : 𝑦 ∈ 𝑉}

where

∗∗

Let us finish this analysis by studying Λ: the composition of the inclusion 𝑗 defined in (35) with the inclusion of M𝑏 (𝑋, 𝑌) into L(H𝑏 (𝑋), H𝑏 (𝑌)). Then the mapping Λ turns out to be holomorphic (endowing L(H𝑏 (𝑋), H𝑏 (𝑌)) with an appropriate topology). We prepare the proof of this fact with a lemma that is a variant of a classical Dunford result; see also [17, Theorem 3].

𝑔 → Λ (𝑔) = Φ𝑔

(51)

is injective and holomorphic if we consider in L(H𝑏 (𝑋), H𝑏 (𝑌)) the 𝜏𝛽 -topology.

Abstract and Applied Analysis

7

Proof. Clearly Λ is well defined. Our first step is to prove that Λ : H𝑏 (𝑌, 𝑋∗∗ ) → L𝜏𝛽 (H𝑏 (𝑋), H𝑏 (𝑌)) is Gˆateaux holomorphic. Let 𝑔1 , 𝑔2 ∈ H𝑏 (𝑌, 𝑋∗∗ ), 𝑓 ∈ H𝑏 (𝑋), 𝑦 ∈ 𝑌, and 𝑡 ∈ C. Here we have Λ(𝑔1 + 𝑡𝑔2 )(𝑓)(𝑦) = 𝑓(𝑔1 (𝑦) + 𝑡𝑔2 (𝑦)). If ∑∞ 𝑚=0 𝑃𝑚 𝑓 is the Taylor series expansion of 𝑓 at 0 on 𝑋, then ∞

∗∗

∗∗

𝑓 (𝑥 ) = ∑ 𝑃𝑚 𝑓 (𝑥 ) ,

and its sum is again 𝑆/(𝑆 − 𝑒(1 + 𝑠0 )𝑀). On the other hand, by using the polarization formula (4), [9, Theorem 3], and Cauchy’s inequalities (6) we get ∨ (𝑚−𝑘) (𝑘) , 𝑔2 (𝑦) ) sup 𝑃𝑚 𝑓 (𝑔1 (𝑦) ‖𝑦‖≤𝑅 𝑚−𝑘 𝑘 ≤ sup 𝑒𝑚 𝑃𝑚 𝑓 𝑔1 (𝑦) 𝑔2 (𝑦) ‖𝑦‖≤𝑅

(52)

𝑚=0

for every 𝑥∗∗ ∈ 𝑋∗∗ and the convergence is absolute and uniform on the bounded subsets of 𝑋∗∗ [8, 9]. Thus

By applying now (55) we obtain ∨ 𝑚 (𝑚−𝑘) (𝑘) 𝑘 , 𝑔2 (𝑦) ) 𝑠0 ∑ ∑ ( ) sup 𝑃𝑚 𝑓 (𝑔1 (𝑦) 𝑘 𝑦 ≤𝑅 𝑘=0 𝑚=𝑘 ‖ ‖

∞

= ∑ 𝑃𝑚 𝑓 (𝑔1 (𝑦) + 𝑡𝑔2 (𝑦))

𝑆 ≤ . 𝑓 𝑆 − 𝑒 (1 + 𝑠0 ) 𝑀 𝐵𝑋 (0,𝑆)

𝑚=0

∨

𝑚

𝑚 (𝑚−𝑘) (𝑘) = ∑ ∑ ( ) 𝑃𝑚 𝑓 (𝑔1 (𝑦) , (𝑡𝑔2 (𝑦)) ) 𝑘 𝑚=0 𝑘=0

∞

𝑒𝑀 𝑚 ) 𝑓𝐵𝑋 (0,𝑆) . 𝑆

∞ ∞

Λ (𝑔1 + 𝑡𝑔2 ) (𝑓) (𝑦)

∞

≤(

(53)

∨

𝑚

Since the function 𝑦 ∈ 𝑌 → 𝑃𝑚 𝑓 (𝑔1 (𝑦)(𝑚−𝑘) , 𝑔2 (𝑦)(𝑘) ) is the composition of a continuous multilinear mapping and two holomorphic mappings of bounded type, it is holomorphic of bounded type on 𝑌. By using (57), we obtain that the series

𝑘=0

∞

∨

where our last step is simply formal. We are going to concentrate our effort now to show that this formal last equality holds in our setting. If we denote by 𝐵𝑌 (0, 𝑅) the closure of 𝐵𝑌 (0, 𝑅), then 𝑔𝑗 (𝐵𝑌 (0, 𝑅)) is a bounded subset of 𝑋∗∗ for 𝑗 = 1, 2. Thus there exists 𝑀 > 0 such that ‖𝑔𝑗 (𝑦)‖ ≤ 𝑀, for every 𝑦 in 𝐵𝑌 (0, 𝑅) and 𝑗 = 1, 2. We fix 𝑠0 > 1 and we take 𝑆 > 0 such that (1 + 𝑠0 )𝑒𝑀 < 𝑆. We have ∞ 𝑚 𝑚 𝑀 𝑚−𝑘 𝑠 𝑀 𝑘 ∑ ∑ 𝑒𝑚 ( ) ( ) ( 0 ) 𝑘 𝑆 𝑆 𝑚=0

=

𝜏𝑏 -converges in H𝑏 (𝑌); hence 𝑇𝑘 (𝑓) belongs to H𝑏 (𝑌) for every 𝑓 in H𝑏 (𝑋). Actually, if we consider 𝑇𝑘 : H𝑏 (𝑋) → H𝑏 (𝑌), this is a linear operator. By (57), it is also continuous, since given 𝑅 > 0 there exists 𝑆 > 0 such that 𝑆 sup 𝑇𝑘 (𝑓) (𝑦) ≤ 𝑓𝐵𝑋 (0,𝑆) , 𝑆 − 𝑒 (1 + 𝑠 ) 𝑀 0 ‖𝑦‖≤𝑅

(59)

for every 𝑓 ∈ H𝑏 (𝑋). Then 𝑞 (𝑇𝑘 ) = sup sup 𝑇𝑘 (𝑓) (𝑦) ‖𝑦‖≤𝑅 𝑓∈B ∞

𝑚

𝑚 𝑒 (1 + 𝑀) ≤ ∑ ( )( ) sup 𝑓𝐵𝑋 (0,𝑆) . 𝑘 𝑆 𝑓∈B

𝑘=0

𝑚=0

(58)

𝑚=𝑘

𝑘=0𝑚=𝑘

∞

∨

𝑇𝑘 (𝑓) := ∑ 𝑃𝑚 𝑓 (𝑔1 (⋅)(𝑚−𝑘) , 𝑔2 (⋅)(𝑘) )

𝑚 (𝑚−𝑘) (𝑘) = ∑ ∑ ( ) 𝑃𝑚 𝑓 (𝑔1 (𝑦) , 𝑔2 (𝑦) ) 𝑡𝑘 , 𝑘

= ∑(

(57)

∨

𝑚 (𝑚−𝑘) (𝑘) = ∑ ∑ ( ) 𝑡𝑘 𝑃𝑚 𝑓 (𝑔1 (𝑦) , 𝑔2 (𝑦) ) 𝑘 𝑚=0 ∞ ∞

(56)

(60)

𝑚=𝑘

𝑚

𝑒 (1 + 𝑠0 ) 𝑀 ) 𝑆

(54)

We have, for 𝑡 ∈ C with |𝑡| ≤ 𝑠0 , again by (57), ∞

𝑆 . 𝑆 − 𝑒 (1 + 𝑠0 ) 𝑀

sup ∑ 𝑞 (𝑇𝑘 ) |𝑡|𝑘

|𝑡|≤𝑠0 𝑘=0

∞

Hence, by the properties of summability of double series of nonnegative numbers, the double series below is convergent in R:

= ∑ 𝑞 (𝑇𝑘 ) 𝑠0𝑘 ≤ 𝑘=0

𝑆 𝑆 − 𝑒 (1 + 𝑠0 ) 𝑀

(61)

× sup 𝑓𝐵𝑋 (0,𝑆) . 𝑓∈B

∞ ∞

𝑚 𝑀 ∑ ∑ 𝑒𝑚 ( ) ( ) 𝑘 𝑆

𝑘=0 𝑚=𝑘

𝑚−𝑘

(

𝑘

𝑠0 𝑀 ) < ∞, 𝑆

(55)

𝑘 As a consequence the series ∑∞ 𝑘=0 𝑇𝑘 𝑡 defined on C with values in L(H𝑏 (𝑋), H𝑏 (𝑌)), 𝜏𝛽 -converges uniformly on the

8

Abstract and Applied Analysis

compacts of C. Hence it is an entire function. Since we have proved that all series involved converge absolutely, we can apply the reordering of absolutely convergent double series to conclude that the last formal equality of (53) actually holds and then ∞

Λ (𝑔1 + 𝑡𝑔2 ) (𝑓) (𝑦) = ∑ 𝑇𝑘 𝑡𝑘

(62)

𝑘=0

is an entire function on C for every 𝑔1 , 𝑔2 . This gives that Λ is Gˆateaux holomorphic. We fix now 𝑞 = 𝑞𝑅,B , a continuous seminorm of the fundamental system defined in (50) and denote by 𝑍𝑞 the completion of the normed space (L(H𝑏 (𝑋), H𝑏 (𝑌))/ Ker 𝑞, 𝑞̂). Given 𝑦 ∈ 𝑌 and 𝑓 ∈ H𝑏 (𝑋) we define the continuous linear functional 𝛿𝑓,𝑦 : L (H𝑏 (𝑋) , H𝑏 (𝑌)) → C

(63)

by 𝛿𝑓,𝑦 (𝑇) = 𝑇(𝑓)(𝑦). Clearly the quotient mapping 𝛿̂𝑓,𝑦 : ̂ = 𝑇(𝑓)(𝑦), belongs to 𝑍∗ . On 𝑍𝑞 → C, defined by 𝛿̂𝑓,𝑦 (𝑇) 𝑞 the other hand the set {𝛿̂𝑓,𝑦 : ‖𝑦‖ ≤ 𝑅, 𝑓 ∈ B} is a norming subset of 𝑍𝑞 since 𝑞 (𝑇) = sup {𝛿𝑓,𝑦 (𝑇) : 𝑦 ∈ 𝑌, 𝑦 ≤ 𝑅, 𝑓 ∈ B} .

(64)

̂ : H𝑏 (𝑌, 𝑋∗∗ ) → 𝑍𝑞 that remains We can consider Λ ̂ : H𝑏 (𝑌, Gˆateaux holomorphic. Thus 𝛿𝑓,𝑦 ∘ Λ = 𝛿̂𝑓,𝑦 ∘ Λ ∗∗ 𝑋 ) → C is Gˆateaux holomorphic for every 𝑦 ∈ 𝑌 and every 𝑓 ∈ H𝑏 (𝑋). If we show that it is continuous, then it will be holomorphic and, by Lemma 6, we will get that ̂ : H𝑏 (𝑌, 𝑋∗∗ ) → 𝑍𝑞 is holomorphic for every seminorm Λ 𝑞. Since L𝜏𝛽 (H𝑏 (𝑋), H𝑏 (𝑌)) is a complete space, we can conclude that Λ : H𝑏 (𝑌, 𝑋∗∗ ) → L𝜏𝛽 (H𝑏 (𝑋), H𝑏 (𝑌)) is holomorphic. Let 𝑔 ∈ H𝑏 (𝑌, 𝑋∗∗ ). Now for fixed 𝑓 ∈ H𝑏 (𝑋) and 𝑦0 ∈ 𝑌, we consider 𝑅 > ‖𝑦0 ‖ and we choose 𝑆 > 0 such that 𝑔 (𝐵𝑌 (0, 𝑅)) ⊂ 𝐵𝑋∗∗ (0, 𝑆) .

(65)

As 𝑓 is uniformly continuous on bounded subsets of 𝑋∗∗ , for a given 𝜀 > 0, there exists 0 < 𝛿 < 1 such that if 𝑧1 , 𝑧2 ∈ 𝐵𝑋∗∗ (0, 𝑆 + 1) with ‖𝑧1 − 𝑧2 ‖ < 𝛿, then 𝑓 (𝑧1 ) − 𝑓 (𝑧2 ) < 𝜀.

(66)

Let ℎ ∈ H𝑏 (𝑌, 𝑋∗∗ ) with sup‖𝑦‖ 0, where Φ𝑢 (𝑓) (𝑦) = Φ (𝜏𝑢∗∗ 𝐽𝑌 𝑦 (𝑓)) (𝑦) = Φ [𝑥 → 𝑓 (𝐽𝑋 𝑥 + 𝑢∗ 𝐽𝑌 𝑦)] (𝑦) ,

(70)

for all 𝑓 ∈ H𝑏 (𝑋) and 𝑦 ∈ 𝐵𝑌 . The fact that Φ𝑢 (𝑓) is in H∞ (𝐵𝑌 ) follows from a similar argument to that in (18) taking 𝑅 = 1. Now, as in (24), we can define a Gelfand transform of 𝑓 ∈ H𝑏 (𝑋) by 𝑓̂ : M𝑏,∞ (𝑋, 𝐵𝑌 ) → H∞ (𝐵𝑌 ) , Φ → 𝑓̂ (Φ) = Φ (𝑓) ,

(71)

and we can see that this is a holomorphic extension of 𝑓 to M𝑏,∞ (𝑋, 𝐵𝑌 ). Proposition 10. Let 𝑋 be a symmetrically regular Banach space and let 𝑌 be any Banach space. Given a function 𝑓 ∈ H𝑏 (𝑋) consider its extension 𝑓̂ defined in (71). Then the restriction of 𝑓̂ to each connected component of M𝑏,∞ (𝑋, 𝐵𝑌 ) is a holomorphic function of bounded type.

Abstract and Applied Analysis

9 ∞

‖Φ (ℎ)‖ = sup |Φ (ℎ) (𝑧)| ≤ sup |ℎ (𝑥)| , ‖𝑧‖ 0 there exists 𝛿 > 0 such that if 𝑧1 , 𝑧2 are in 𝐵𝑋∗∗ (0, 𝑅 + 𝑀) with ‖𝑧1 − 𝑧2 ‖ < 𝛿, then |𝑓(𝑧1 ) − 𝑓(𝑧2 )| < 𝜀. Consider now 𝑢1 , 𝑢2 in L(𝑋∗ , 𝑌∗ ) with ‖𝑢𝑗 ‖ < 𝑀 for 𝑗 = 1, 2 and ‖𝑢1 − 𝑢2 ‖ < 𝛿. We have 𝑇 (𝑢1 ) − 𝑇 (𝑢2 ) = sup Φ𝑢1 (𝑓) (𝑦) − Φ𝑢2 (𝑓) (𝑦) ‖𝑦‖